246 STEP 4. Review the Knowledge You Need to Score High
Example 4
If
∫k
−k
f(x)dx= 2
∫k
0
f(x)dxfor all values ofk, then which of the following could be the
graph off? (See Figure 11.3-1.)
Figure 11.3-1
∫k
−k
f(x)dx=
∫ 0
−k
f(x)dx+
∫k
0
f(x)dx
Since
∫k
−k
f(x)dx= 2
∫k
0
f(x)dx, then
∫k
0
f(x)dx=
∫ 0
−k
f(x)dx.
Thus,fis an even function. Choice (C).
11.4 Improper Integrals
Main Concepts:Infinite Intervals of Integration, Infinite Discontinuities
Infinite Intervals of Integration
Improper integrals are integrals with infinite intervals of integration or infinite discontinu-
ities within the interval of integration. For infinite intervals of integration,
∫∞
a
f(x)dx=
liml→∞
∫l
a
f(x)dxand
∫b
−∞
f(x)dx=llim→−∞
∫b
1
f(x)dx. If the limit exists, the integral converges.
∫∞
−∞
f(x)dx=
∫c
−∞
f(x)dx+
∫∞
c
f(x)dxfor some valuec.