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



€

Area ≈ f (x i*)Δx

i=1

16

∑

Area


How do we make it exact?€


i=1

16

∑

Area



€


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 .


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

∫


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


(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




€

L(t) =dL

dtdt∫

€

= 6.48e−0.09tdt∫= −72e−0.09t +C

€

L(0) = 0⇒ C = 72

€

∴L(t) = −72e−0.09t + 72

€

L(t) = 45 when t ≈11 years

announcements topics: -sections 7.3 (definite integrals) and 7.4 (ftc) * read these sections and...

Documents

number of rectangles

definite integraldefinition

definite integralestimate

definite integralsexample

signed area

terms of area

arealefthand estimate

riemann sums