407
Answers
601–700
Answers and Explanations
π
π
π
π
π
x
dx
x dx
x
4
2
0
1
8
0
1
9
0
1
9
1
9
0
9
( ) =
=

 

 
=
−
( )
=
∫
∫
663.
2π
Recall the definition of volume: Let S be a solid that lies between y = c and y = d. If A(y)
is the cross-sectional area of S in the plane P y that goes through y and is perpendicular
to the y-axis, where A is a continuous function, then the volume of S is
V
A y dx
c
d
= ∫ ( )
Here’s the region that’s being rotated. Because the cross-sectional slice is a circle, you
have A(y) = π(x)
2
= π(g(y))
2
.
In this case, the limits of integration correspond to y = 0 and y = π. Because
g y
y
( )
sin
=
, the integral to find the area becomes
π
π
π
π
π
π
π
π
π
sin
s in
( cos )
(cos
cos )
(
)
y dy
ydy
y
(
) =
= −
= −
−
= − − −
∫
∫
0
2
0
0
0
1 1
= = 2π