Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and...
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Transcript of Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and...
Announcements
Topics: - sections 7.3 (definite integrals) and 7.4 (FTC)* Read these sections and study solved examples in your
textbook!
Work On:- Practice problems from the textbook and assignments
from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)
Area
How do we calculate the area of some irregular shape?
For example, how do we calculate the area under the graph of f on [a,b]?
€
Area = ?
Area
Approach:
€
x0 =
€
x1
€
x2
€
x3
€
=x4
€
n = 4number of rectangles:
€
Δx =b− a
nwidth of each rectangle:
We approximate the area using rectangles.
AreaLeft-hand estimate: Let the height of each rectangle be given by the valueof the function at the left endpoint of the interval.
€
x0 =
€
x1
€
x2
€
x3
€
=x4
AreaLeft-hand estimate:
€
Area ≈ f (x0)Δx + f (x1)Δx + f (x2)Δx + f (x3)Δx
€
≈( f (x0) + f (x1) + f (x2) + f (x3))Δx
€
≈ f (x i)Δxi= 0
3
∑Riemann Sum
AreaRight-hand estimate: Let the height of each rectangle be given by the valueof the function at the right endpoint of the interval.
€
x0 =
€
x1
€
x2
€
x3
€
=x4
AreaRight-hand estimate:
€
Area ≈ f (x1)Δx + f (x2)Δx + f (x3)Δx + f (x4 )Δx
€
≈( f (x1) + f (x2) + f (x3) + f (x4 ))Δx
€
≈ f (x i)Δxi=1
4
∑Riemann Sum
AreaMidpoint estimate: Let the height of each rectangle be given by the valueof the function at the midpoint of the interval.
€
x1
€
x2
€
x3
€
x4
AreaMidpoint estimate:
€
Area ≈ f (x1*)Δx + f (x2
*)Δx + f (x3*)Δx + f (x4
* )Δx
€
≈ f (x i*)Δx
i=1
4
∑Riemann Sum
€
≈( f (x1*) + f (x2
*) + f (x3*) + f (x4
* ))Δx
Area
How can we improve our estimation?Increase the number of rectangles!!!
How do we make it exact?Let the number of rectangles go to infinity!!!
€
Area ≈ f (ti*)Δt
i=1
16
∑
Area
How can we improve our estimation?Increase the number of rectangles!!!
How do we make it exact?Let the number of rectangles go to infinity!!!
€
Area ≈ f (x i*)Δx
i=1
16
∑
Area
How can we improve our estimation?Increase the number of rectangles!!!
How do we make it exact?€
Area ≈ f (x i*)Δx
i=1
16
∑
Area
How can we improve our estimation?Increase the number of rectangles!!!
How do we make it exact?Let the number of rectangles go to infinity!!!
€
Area ≈ f (x i*)Δx
i=1
16
∑
Riemann Sums and the Definite Integral
Definition:The definite integral of a function on the interval from a to b is defined as a limit of the Riemann sum
where is some sample point in the interval and
€
f (x)dx = limn→∞
f (x i*)Δx
i=1
n
∑a
b
∫ €
f
€
Δx =b− a
n.
€
x i*
€
[x i−1, x i]
The Definite Integral
Interpretation:
If , then the definite integral is the area under the curve from a to b.
€
f ≥ 0
€
Area = f (x)dxa
b
∫
€
y = f (x)
Estimating a Definite Integral
Estimate using left-endpoints,
midpoints, and right-endpoints with n=4.
€
ln xdx1
3
∫
The Definite Integral
Interpretation:
If is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f
€
f
€
f (x)dx−1
4
∫ = net area
Evaluating Definite Integrals
Example:Evaluate the following integrals by interpreting each in terms of area.
(a)(b)
(c) €
(x −1) dx0
3
∫
€
1 − x 2 dx0
1
∫
€
sin x dx−π
π
∫
Properties of Integrals
Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.
€
(1) f (x)dx = 0a
a
∫
€
(2) f (x)dx = −a
b
∫ f (x)dxb
a
∫
€
(3) c f (x)dx =a
b
∫ c f (x)dxa
b
∫
Properties of Integrals
Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.
€
(4) f (x)± g(x)( )dx = f (x)dx ± g(x)dxa
b
∫a
b
∫a
b
∫
€
(5) c dx = c(b− a)a
b
∫
Summation Property of the Definite Integral
(6) Suppose f(x) is continuous on the interval from a to b and that
Then
€
a ≤ c ≤ b.
€
f (x)dxa
b
∫ = f (x)dxa
c
∫ + f (x)dxc
b
∫ .€
f (x)dxa
c
∫
€
f (x)dxc
b
∫
Properties of the Definite Integral
(7) Suppose f(x) is continuous on the interval from a to b and that
Then
€
m ≤ f (x) ≤ M .
€
m(b− a) ≤ f (x)dxa
b
∫ ≤ M(b− a).
Types of Integrals
• Indefinite Integral
• Definite Integral€
f (x)dx = F(x) +C∫antiderivative of f
function of x
€
f (x)dx = net areaa
b
∫number
The Fundamental Theorem of Calculus
If is continuous on then
where is any antiderivative of , i.e.,
€
f (x)dxa
b
∫ = F(x) ab = F(b) − F(a)
€
f
€
[a, b],
€
F
€
f
€
F '= f .
Evaluating Definite Integrals
Example:Evaluate each definite integral using the FTC.
(a) (b)
(c) (d) €
(x −1)dx0
3
∫
€
(4
t+t
4)dt
1
2
∫
€
1
1− 4x 2dx
− 14
12
∫
€
(3x −1)2
xdx
1
2
∫
Evaluating Definite Integrals
Example:Try to evaluate the following definite integral using the FTC. What is the problem?
€
1
(x − 2)2 dx1
4
∫
Differentiation and Integration as Inverse Processes
If f is integrated and then differentiated, we arrive back at the original function f.
If F is differentiated and then integrated, we arrive back at the original function F.
€
d
dxF(x)dx = F(x) a
b
a
b
∫€
d
dxf (t)dt
a
x
∫ = f (x)
FTC II
FTC I
The Definite Integral - Total Change
Interpretation:
The definite integral represents the total amount of change during some period of time.
Total change in F between times a and b:
€
F(b) − F(a) =dF
dtdt
a
b
∫value at end value at start
rate of change
Application – Total Change
Example: Suppose that the growth rate of a fish is given by the differential equation
where t is measured in years and L is measured in centimetres and the fish was 0.0 cm at age t=0 (time measured from fertilization).€
dL
dt= 6.48e−0.09t
Application – Total Change
(a) Determine the amount the fish grows between 2 and 5 years of age.
(b) At approximately what age will the fish reach 45cm?
Application – Total Change
(a) Determine the amount the fish grows between 2 and 5 years of age.
(b) At approximately what age will the fish reach 45cm?
€
L(5) − L(2) =dL
dtdt
2
5
∫
€
= 6.48e−0.09tdt2
5
∫
€
= −72[ e−0.09t
2
5
= −72e−0.09(5)[ ] − −72e−0.09(2)
[ ]
≈14.2 cm