First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.
-
Upload
ami-joseph -
Category
Documents
-
view
217 -
download
2
Transcript of First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.
![Page 1: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/1.jpg)
First Fundamental Theorem of Calculus
Greg Kelly, Hanford High School, Richland, Washington
![Page 2: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/2.jpg)
When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.
211
8V t
subinterval
partition
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
Subintervals do not all have to be the same size.
![Page 3: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/3.jpg)
211
8V t
subinterval
partition
If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .P
As gets smaller, the approximation for the area gets better.
P
0
1
Area limn
i iP
i
f c x
if P is a partition of the interval ,a b
![Page 4: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/4.jpg)
0
1
limn
i iP
i
f c x
is called the definite integral of
over .f ,a b
If we use subintervals of equal length, then the length of a
subinterval is:b a
xn
The definite integral is then given by:
1
limn
in
i
f c x
![Page 5: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/5.jpg)
1
limn
in
i
f c x
Leibnitz introduced a simpler notation for the definite integral:
1
limn b
i ani
f c x f x dx
Note that the very small change in x becomes dx.
![Page 6: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/6.jpg)
b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrandvariable of integration
(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
![Page 7: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/7.jpg)
b
af x dx
We have the notation for integration, but we still need to learn how to evaluate the integral.
![Page 8: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/8.jpg)
time
velocity
After 4 seconds, the object has gone 12 feet.
Let’s consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance: 3t d
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
ft3 4 sec 12 ft
sec
![Page 9: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/9.jpg)
If the velocity varies:
11
2v t
Distance:21
4s t t
(C=0 since s=0 at t=0)
After 4 seconds:1
16 44
s
8s
1Area 1 3 4 8
2
The distance is still equal to the area under the curve!
Notice that the area is a trapezoid.
![Page 10: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/10.jpg)
211
8v t What if:
We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.
It seems reasonable that the distance will equal the area under the curve.
![Page 11: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/11.jpg)
We can use anti-derivatives to find the area under a curve!
![Page 12: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/12.jpg)
Fundamental Theorem of Calculus
b
af x dx F b F a
Just like we proved earlier!
Area under curve from a to x = antiderivative at x minus
antiderivative at a.
![Page 13: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/13.jpg)
Area from x=0to x=1
Example: 2y x
Find the area under the curve from x=1 to x=2.
2 2
1x dx
2
1
31
3x
3 313
21 1
3
8 1
3 3
7
3
Area from x=0to x=2
Area under the curve from x=1 to x=2.
![Page 14: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/14.jpg)
Example: 2y x
Find the area under the curve from x=1 to x=2.
2 , , 1, 2x x
To use your TI-83+ or TI-84+
Math
7. fnInt Enter
Function
VariableLimits of Integration
![Page 15: First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.](https://reader036.fdocuments.in/reader036/viewer/2022082611/56649ed05503460f94bde110/html5/thumbnails/15.jpg)
Example:
Find the area between the
x-axis and the curve
from to .
cosy x
0x 3
2x
2
3
2
3
2 2
02
cos cos x dx x dx
/ 2 3 / 2
0 / 2sin sinx x
3sin sin 0 sin sin
2 2 2
1 0 1 1
3
pos.
neg.