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