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IntegralsIntegralsMAC 2233

A function F is an antiderivative of f on an interval I if _______________ for every x in I.

Antiderivatives

If G is an antiderivative of f, then every antiderivative of f must have the form

where C is ___________

The process of finding all antiderivatives of a function is called antidifferentiation or integration

Indefinite Integrals

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Rules of Integration

, where , are constantsk dx kx C k C= +∫1

∫ 11 , where 11

n nx dx x C nn

+= + ≠ −+∫

( ) ( )cf x dx c f x dx=∫ ∫

[ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx± = ±∫ ∫ ∫

Rules of Integration

1 ln | |dx x Cx

= +∫

x xe dx e C= +∫

Example

• Integrate 4 dx−∫

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Example

• Integrate 21 3 6x x dx+ −∫

Example

• Integrate 1.7 2.5x x dx−−∫

Example

• Integrate

• Rewrite

13

1 2x dxx

−+∫• Rewrite

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Example

• Integrate

• Rewrite

59 x dx∫• Rewrite

Example

• Integrate

• Rewrite

0.4

0.4

4.2 23

xx e dxx

+ −∫• Rewrite

Homework

• p. 381 problems 1-29 odd, 37, 43, 45

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How do we integrate ?

1. Let u = g (x), where g is part of the integrand, usually the of the composite

Substitution2 3 73 ( 1)x x dx+∫

usually the _______________ of the composite function f (g (x)).

2. Compute ______________.3. Use the substitution __________________ to

convert the entire integral into one involving only u.

4. Evaluate the resulting integral.5. Replace u by g (x) to obtain the final solution as a

function of x.

Example

• Integrate3

2 3 23 ( 2)t t dt+∫

Example

• Integrate2

3 2

3 2( 2 )

x dxx x

++∫

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Example

• Integrate ∫ + dxxx 743 )9(

3 2x dx

x +∫Example

• Integrate

17 5

dxx−∫

Example

• Integrate

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2xxe dx∫Example

• Integrate

Example

The current circulation of the Investor’s Digest is 3000 copies per week. The managing editor of the weekly projects a growth rate of weekly projects a growth rate of

copies per week, t weeks from now, for the next 3 years. Based on her projection, what will the circulation of the digest be 125 weeks from now?

From Calculus for the Managerial, Life, and Social Sciences, 6th ed. By Tan, 2003, example 12, p.406.

Homework

• p. 394 problems 3-35 odd, 45, 51, 55, 61, 67

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How do we calculate the area of the region bounded by the graph of a nonnegative function, f, the x-axis, and the vertical lines x = a and x = b?

Area Under the Curve

Let f be a nonnegative, continuous function on [a, b]. Then the area of the region under the graph of f is

Area under the curve

graph of f is

Let f be a continuous function defined on [a, b]. If

The Definite Integral

exists for all choices of x1, …, xn in the subintervals of [a, b] then this limit is called the definite integral of f from a to b and we write

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Properties of the Definite Integral

( ) ( )b b

a a

kf x dx k f x dx=∫ ∫

[ ( ) ( )] ( ) ( )b b b

f x g x dx f x dx g x dx± ±∫ ∫ ∫[ ( ) ( )] ( ) ( )a a a

f x g x dx f x dx g x dx± = ±∫ ∫ ∫

( ) ( ) ( )b c b

a a c

f x dx f x dx f x dx= +∫ ∫ ∫

( ) 0a

a

f x dx =∫

Let f be a continuous function on [a, b]. Then

The Fundamental Theorem of Calculus

where F is any antiderivative of f ; that is F’(x) = f (x). We write

Example

• Find the area of the region under f (x) = 4x – 1 on the interval [2, 4].

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Example

• Evaluate

0

1

4 x dx−

−∫

• Evaluate

Example2

5 3

1

1t t dt− +∫

Example

• Evaluate1

2 2

0

3 ( 1)x x dx−∫

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Example

• Evaluate0 3

4 41 (2 )

t dtt− −∫

Example

• Evaluate

0

1

14 5

dxx− −∫

Net Change

• The definite integral represents the net change in the antiderivative function

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Example

A certain oil well that yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $95 per barrel and is crude oil is currently $95 per barrel and is expected to rise at a constant rate of 30 cents per barrel per month. If the oil is sold as soon as it is extracted from the ground, what will be the total future revenue from the well?

From Calculus for Business, Economics and the Social and Life Sciences, 10th ed. By Hoffmann & Bradley, 2007, problem 50, p.412.

Homework

• p. 410 problems 1-29 odd, 41, 43, 45, 49, 53, 55, 57 59 6357, 59, 63