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Transcript of The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the...
![Page 1: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/1.jpg)
The FTC Part 2, Total Change/Area & U-Sub
![Page 2: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/2.jpg)
Question from Test 1
Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow?
A. log(7)/3 B. (1/3)log(13/7) C. 3 log (13/7) D. 3log(7) E. Never
![Page 3: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/3.jpg)
Question from Test 1
Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow?
A. log(7)/3
![Page 4: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/4.jpg)
The Total Change Theorem
′F x( )a
b
∫ dx = F b( ) − F a( )
The integral of a rate of change is the total change from a to b. (displacement)
(still from last weeks notes)
![Page 5: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/5.jpg)
The Total Change Theorem Ex: Given
Find the displacement and total distance traveled from time 1 to time 6. Displacement: (negative area takes away from positive)
Total Distance: (all area counted positive)
v(t) =t2 −4t+ 3
t 2 −4t+ 3( )dt1
6
∫
− t 2 − 4t + 3( )dt1
3
∫ + t 2 − 4t + 3( )dt3
6
∫
![Page 6: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/6.jpg)
Total Area Find the area of the region bounded by
the x-axis, y-axis and y = 2 – 2x. First find the bounds by setting 2 – 2x = 0 and by substitution 0 in for x
€
2 − 2x( )dx0
1∫
![Page 7: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/7.jpg)
Total Area Ex. Find the area of the region
bounded by the y-axis and the curve
€
x = 2y2 + 3y4 − 2y6
€
2y2 + 3y4 − 2y6( )0
2∫ dy
![Page 8: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/8.jpg)
Fundamental Theorem of Calculus (Part 1)(Chain Rule)
If f is continuous on [a, b], then the function defined by
is continuous on [a, b] and differentiable on (a, b) and
g(x) = f(t)dta
u(x)
∫ a≤x≤b
g '(x) = f(u(x))u'(x)
![Page 9: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/9.jpg)
Fundamental Theorem of Calculus (Part 1)
d
drv eu2 v −1 dv
1
r3
∫ 3r2( )r3 eu2r3 −1
d
drv eu2 v −1 dv
1
r3
∫
![Page 10: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/10.jpg)
Fundamental Theorem of Calculus (Part 2)
If f is continuous on [a, b], then :
Where F is any antiderivative of f.
( )
f (t)dta
b
∫ =F(b)−F(a)
F ' = f
Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!
![Page 11: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/11.jpg)
Example
![Page 12: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/12.jpg)
Example
We have to • find an antiderivative;• evaluate at 3;• evaluate at 2;• subtract the results.
![Page 13: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/13.jpg)
Example
![Page 14: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/14.jpg)
Example
This notation means: evaluate the function at 3 and 2, and subtract the
results.
![Page 15: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/15.jpg)
Example
![Page 16: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/16.jpg)
Example
![Page 17: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/17.jpg)
Example
![Page 18: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/18.jpg)
Example
Don’t need to include “+ C” in our antiderivative, because any antiderivative
will work.
![Page 19: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/19.jpg)
Example
the “C’s” will cancel each other out.
x3dx2
3
∫ =14x4 +C⎡
⎣⎢⎤⎦⎥2
3
=1
434 + C
⎛⎝⎜
⎞⎠⎟
−1
424 + C
⎛⎝⎜
⎞⎠⎟
=1
434 + C −
1
424 − C
=81
4−
16
4=
65
4
![Page 20: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/20.jpg)
Example
![Page 21: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/21.jpg)
Example
![Page 22: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/22.jpg)
Example
Alternate notation
![Page 23: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/23.jpg)
Example
![Page 24: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/24.jpg)
Example
![Page 25: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/25.jpg)
Example
= –1
![Page 26: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/26.jpg)
Example
= –1 = 1
![Page 27: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/27.jpg)
Example
![Page 28: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/28.jpg)
Example
![Page 29: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/29.jpg)
Example
![Page 30: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/30.jpg)
Given: f (x) =
0 x< 0x 0 ≤x≤12 −x 1 < x≤20 x> 2
⎧
⎨⎪⎪
⎩⎪⎪
Write a similar expression for the continuous function:
g(x) = f (t)dt0
x
∫g(x) =
0 x< 0
x2
2 0 ≤x≤1
2x−x2
2−1 1 < x≤2
1 x> 2
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
![Page 31: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/31.jpg)
Fundamental Theorem of Calculus (Part 2)
If f is continuous on [a, b], then :
Where F is any antiderivative of f. ( )
f (t)dta
b
∫ =F(b)−F(a)
F ' = f
![Page 32: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/32.jpg)
Evaluate:
Multiply out:
2x +1( )2 dx−1
2
∫
= 4x2 + 4x +1( )dx−1
2
∫
Use FTC 2 to Evaluate:
=[4
3x3 + 2x + x
−1
2
=4
3(2)3 + 2(2)2 + 2 − (
4
3(−1)3 + 2(−1)2 −1)
=21
![Page 33: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/33.jpg)
What if instead?
It would be tedious to use the same multiplication strategy!
There is a better way!
We’ll use the chain rule (backwards)
2x +1( )10 dx∫
![Page 34: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/34.jpg)
Chain Rule for Derivatives:
d
dxf g x( )( )⎡⎣ ⎤⎦= f ' g(x)( )g'(x)
Chain Rule backwards for Integration:
f ' g(x)( )g'(x)dx=∫ f g x( )( ) +C
![Page 35: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/35.jpg)
Look for:
f ' g(x)( ) g'(x)dx=∫ f g x( )( ) +C
![Page 36: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/36.jpg)
Back to Our Example
Let
2x +1( )2 dx−1
2
∫
2x +1( )2dx
−1
2
∫u =2x+1du =2dx
=1
22x +1( )
22dx
−1
2
∫2x +1( )2 dx−1
2
∫
![Page 37: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/37.jpg)
Our Example as anIndefinite Integral
With
AND Without worrying about the bounds for now:
Back to x (Indefinite):
2x +1( )2 dx∫
u =2x+1du =2dx
1
22x +1( )2 2dx∫
=1
62x +1( )
3+ C
=1
2u2du∫ =
1
6u3 + C
![Page 38: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/38.jpg)
The same substitution holds for the higher power!
With
Back to x (Indefinite):
2x +1( )10 dx∫
u =2x+1du =2dx
1
22x +1( )10 2dx∫
=1
222x +1( )
11+ C
=1
2u10du∫ =
1
22u11 + C
![Page 39: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/39.jpg)
Our Original Exampleof a Definite Integral:
To make the substitution complete for a Definite Integral: We make a change of bounds using:
When x = -1, u = 2(-1)+1 = -1 When x = 2, u = 2(2) + 1 = 5
The x-interval [-1,2] is transformed to the u-interval [-1, 5]
2x +1( )2 dx−1
2
∫
u =2x+1
1
22x +1( )2 2dx
−1
2
∫
=16
53 −(−1)3( )
=1
2u2du
−1
5
∫ =1
6u3
−1
5
=21
![Page 40: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/40.jpg)
Substitution Rule for Indefinite Integrals
If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then
f g(x)( )g'(x)dx= f(u)du∫∫Substitution Rule for Definite Integrals
If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then
f g(x)( )g'(x)dxa
b
∫ = f(u)dug(a)
g(b)
∫
![Page 41: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/41.jpg)
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
Evaluate:
![Page 42: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/42.jpg)
Evaluate:
Looks almost like cos(x2) 2x, which is
the derivative of sin(x2).
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
![Page 43: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/43.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
d
dxsin x2( )( ) =2xcos x2( )
Using the Chain Rule, we know that:
![Page 44: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/44.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫
![Page 45: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/45.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫=12
cos x2( )2xdx∫
![Page 46: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/46.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
We put in a 2 so the pattern will match.
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫=12
cos x2( )2xdx∫
![Page 47: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/47.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
We put in a 2 so the pattern will match.
So we must also put in a 1/2 to keep the
problem the same.
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫=12
cos x2( )2xdx∫
![Page 48: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/48.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫=12
cos x2( )2xdx∫
![Page 49: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/49.jpg)
Evaluate:
We will rearrange the integral to get an exact match:
Using the Chain Rule, we know that:
d
dxsin x2( )( ) =2xcos x2( )
x cos x2( )dx= cos x2( )xdx∫∫=12
cos x2( )2xdx∫
=12sin x2( )⎡⎣ ⎤⎦+C
![Page 50: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/50.jpg)
Check Answer:
![Page 51: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/51.jpg)
Check:
Check Answer:
![Page 52: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/52.jpg)
Check:
From the chain rule
Check Answer:
![Page 53: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/53.jpg)
1) Choose u.
Indefinite Integrals by Substitution
![Page 54: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/54.jpg)
1) Choose u.
2) Calculate du.
€
du =du
dxdx
Indefinite Integrals by Substitution
![Page 55: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/55.jpg)
1) Choose u.
2) Calculate du.
3) Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!)€
du =du
dxdx
Indefinite Integrals by Substitution
![Page 56: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/56.jpg)
1) Choose u.
2) Calculate du.
3) Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!)
4) Solve the new integral.
€
du =du
dxdx
Indefinite Integrals by Substitution
![Page 57: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/57.jpg)
Indefinite Integrals by Substitution
1) Choose u.
2) Calculate du.
3) Substitute u.Arrange to have du in your integral also.(All xs and dxs must be replaced!)
4) Solve the new integral.
5) Substitute back in to get x again.
€
du =du
dxdx
![Page 58: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/58.jpg)
ExampleA linear substitution:
Let u = 3x + 2. Then du = 3dx.
e3x+2dx∫ =13
e3x+23dx∫
=1
3eudu∫
=1
3eu + C
=1
3e3x+2 + C
![Page 59: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/59.jpg)
Choosing u Try to choose u to be an inside function.
(Think chain rule.) Try to choose u so that du is in the
problem, except for a constant multiple.
![Page 60: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/60.jpg)
Choosing uFor
u = 3x + 2 was a good choice because
(1) 3x + 2 is inside the exponential.
(2) The derivative is 3, which is only a constant.
![Page 61: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/61.jpg)
PracticeLet u =
du =
![Page 62: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/62.jpg)
PracticeLet u = x2 + 1
du =
![Page 63: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/63.jpg)
PracticeLet u = x2 + 1
du = 2x dx
![Page 64: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/64.jpg)
PracticeLet u = x2 + 1
du = 2x dx
![Page 65: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/65.jpg)
PracticeLet u = x2 + 1
du = 2x dx
Make this a 2x dx and we’re
set!
![Page 66: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/66.jpg)
PracticeLet u = x2 + 1
du = 2x dx
![Page 67: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/67.jpg)
PracticeLet u = x2 + 1
du = 2x dx
![Page 68: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/68.jpg)
PracticeLet u = x2 + 1
du = 2x dx
![Page 69: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/69.jpg)
PracticeLet u = x2 + 1
du = 2x dx
= 1
2ln x2 + 1 + C
![Page 70: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/70.jpg)
PracticeLet u =
du =
![Page 71: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/71.jpg)
PracticeLet u = sin(x)
du =
![Page 72: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/72.jpg)
PracticeLet u = sin(x)
du = cos(x) dx
![Page 73: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/73.jpg)
PracticeLet u = sin(x)
du = cos(x) dx
![Page 74: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/74.jpg)
PracticeLet u = sin(x)
du = cos(x) dx
![Page 75: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/75.jpg)
PracticeLet u = sin(x)
du = cos(x) dx
![Page 76: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/76.jpg)
PracticeLet u =
du = An alternate possibility:
![Page 77: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/77.jpg)
PracticeLet u = cos(x)
du = –sin(x) dxAn alternate possibility:
![Page 78: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/78.jpg)
PracticeLet u = cos(x)
du = –sin(x) dxAn alternate possibility:
![Page 79: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/79.jpg)
PracticeLet u = cos(x)
du = –sin(x) dxAn alternate possibility:
![Page 80: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/80.jpg)
PracticeLet u = cos(x)
du = –sin(x) dxAn alternate possibility:
![Page 81: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/81.jpg)
Practice
Note:
![Page 82: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/82.jpg)
Practice
Note:
![Page 83: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/83.jpg)
Practice
Note:
What’s the difference?
![Page 84: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/84.jpg)
PracticeNote:
What’s the difference?
![Page 85: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/85.jpg)
Practice
Note:
What’s the difference?
![Page 86: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/86.jpg)
Practice
Note:
What’s the difference?
This is 1!
![Page 87: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/87.jpg)
Practice
Note:
What’s the difference?
![Page 88: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/88.jpg)
Practice
Note:
What’s the difference?
That is, the difference is a constant.
![Page 89: The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.](https://reader036.fdocuments.in/reader036/viewer/2022070407/56649e3f5503460f94b2fb70/html5/thumbnails/89.jpg)
In-Class Assignment
Integrate using two different methods:
1st by multiplying out and integrating 2nd by u-substitution
Do you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!)
If you don’t get exactly the same answer, is it a problem? Why or why not?
3x −1( )∫2dx