5 Steps to a 5 AP Calculus BC 2019

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.