4/6/2010 1 Integrals Integrals MAC 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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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.