5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 259

Example 1

IfF(x)=

∫x

0

2 costdtfor 0≤x≤ 2 π, find the value(s) ofxwherefhas a local minimum.

Method 1: Sincef(x)=

∫x

0

2 costdt, f′(x)=2 cosx.

Setf′(x)=0; 2 cosx=0,x=

π

2

or

3 π

2

.

f′′(x)=−2 sinxand f′′

(

π

2

)

=−2 and f′′

(

3 π

2

)

=2.

Thus, atx=

3 π

2

, fhas a local minimum.

Method 2: You can solve this problem geometrically by using area. See Figure 12.1-3.

[0,2π] by [−3,3]

Figure 12.1-3

The area “under the curve” is above thet-axis on[0,π/ 2 ]and below thex-axis

on[π/2, 3π/ 2 ]. Thus the local minimum occurs at 3π/2.

Example 2

Letp(x)=

∫x

0

f(t)dtand the graph of fis shown in Figure 12.1-4.

t

y

f(t)

(^0) 12 345678
–4
4
Figure 12.1-4