Chapter 6
Applications of Integration
Math 1B
§ 6.1 Areas Between Curves
Overview: In Chapter 5 we found areas of regions that lie under the graphs of functions. In this section,
we use integrals to find the area between the graphs of two functions.
Before we begin, recall the following definition.
Definition of a Definite Integral: If f is a function defined for a ≤ x ≤ b , we divide the interval [ a,b]
into n subintervals of equal width Δ x = (b − a) /n . We let x 0 (= a) , x1, x 2 , … , x n (= b) be the endpoints
of these subintervals and we let x1* , x *2 , … , x *n be any sample points in these subintervals, so x *i lies in
the ith subinterval [ x i−1, x i ] . Then the definite integral of
€ f from a to b is
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€
€
€
€
b
€
∫
a
n
f ( x )€
dx =€lim ∑ f€( xi* ) Δx
n→∞
€
i=1
provided that this limit exists.
Areas Between Curves
Suppose f ( x ) and g( x ) are continuous on [ a,b] and g( x ) ≤ f ( x ) for each x in [ a,b] .
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Divide this region into n strips of equal width. Then the area of the ith strip can be approximated by a
rectangle with base Δx and height f ( x *i ) − g( x *i ) .
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Stewart-7e
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1
The area of this region can be approximated by the Riemann sum
n
∑ [ f ( x ) − g( x )] Δx
*
i
*
i
i=1
The more rectangles we have, the better the approximation. (i.e. as n → ∞ )
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So we define the area of this region as
n
€
[
A = lim ∑ f ( x ) − g( x
n →∞
i=1
*
i
*
i
)] Δx
But this is just the definite integral of f − g .
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Area Formula: The area A of the region bounded above by y = f ( x ) , below by y = g( x ) , on the left
by x = a, and on the right by x = b , where f and g are continuous and g( x ) ≤ f ( x ) for each x in [ a,b] ,
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is
b
A=
€
∫ [ f ( x ) − g( x )] dx
€
€a
€
€
€
Example: Find the area of the region bounded by y = x and y = 6 − x 2 .
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€
Stewart – 7e
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2
Example: Find the area of the region bounded by y = cos x and y = 1− cos x , 0 ≤ x ≤ π .
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Stewart – 7e
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3
Sometimes it is easier to treat x as a function of y.
The area A of the region bounded by x = f ( y ) , x = g( y ) , y = c , and y = d where f and g are
continuous and g( y ) ≤ f ( y ) for each y in [c,d ] , is
d
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€
€
A = ∫ "# f ( y) €
− g ( y)$% dy€
c
€
Example: Find the area of the region bounded by x = 2y 2 and x = 4 + y 2 .
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Stewart – 7e
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4