413
Answers
601–700
Answers and Explanations
π
π
π
π
π
y
dy
y dy
y
2 3
0
8
2
4 3
0
8
7 3
0
8
7 3
3
7
3
7
8
0
384
7
( ) =
=
=
−
(
)
=
∫
∫
672.
4
3
3
π r
Recall the definition of volume: Let S be a solid that lies between x = a and x = b. If A(x)
is the cross-sectional area of S in the plane P x that goes through x and is perpendicular
to the x-axis, where A is a continuous function, then the volume of S is
V
A x dx
a
b
= ∫ ( )
Because the cross-sectional slice is a circle in this case, you have A(x) = π(y)
2
= π(f
 
(x))
2
.
Notice that y
r
x
=
−
2
2 intersects y = 0 when x = r and when x = –r, which corresponds
to the limits of integration. Therefore, the integral to find the volume becomes
π
π
r
x
dx
r
x dx
r
r
r
r
2
2
2
2
2
−
(
) =
−
(
)
−
−
∫
∫
Because y = r
2
– x
2
is an even function, you can change the lower limit of integration to
zero and multiply by 2 to evaluate the integral:
π
π
π
π
r
x dx
r
x dx
r x x
r r
r
r
r
r
r
2
2
2
2
0
2
3
0
2
3
2
2
3
2
−
(
)
=
−
(
)
=
−






=
−
−
∫
∫
( ) 3 3
4
3
3






= π r
By rotating the semicircular region, you’ve found the formula for the volume of a
sphere!
673.
π
Recall the definition of volume: Let S be a solid that lies between x = a and x = b. If A(x)
is the cross-sectional area of S in the plane P x that goes through x and is perpendicular
to the x-axis, where A is a continuous function, then the volume of S is
V
A x dx
a
b
= ∫ ( )