Part II: The Answers
414
Answers
601–700
Here’s the region that’s being rotated:
In this case, the cross-sectional slice has the shape of a washer. When the cross-sectional
slice is a washer, you can find A(x) by using
A
r
r
=
= ( )
( )
π
π
π
(
)
(
)
outer radius
inner radius
out
i n
2
2
2
2
−
− π
where the outer radius, r out , is the distance of the function farther away from the line of
rotation and the inner radius, r in , is the distance of the function closer to the line of
rotation at a particular value of x.
Begin by finding the point of intersection of the two functions by setting them equal to
each other: sin x = cos x, which has the solution x = π
4
on the interval 0 2
, π
( ) . Notice
that on the interval 0 4
, π
( ) , you have cos x > sin x so that r out = cos x and r in = sin x, and
on the interval π π
4 2
,
( ) , you have sin x > cos x so that r out = sin x and r in = sin x.
Therefore, the integrals to find the volume are
π
π
π
π
π
(cos ) (sin )
(sin ) (cos )
x
x dx
x
x dx
2
2
0
4
2
2
4
2
−
+
−
∫
∫
Because π
π (cos ) (sin )
x
x dx
2
2
0
4
−
∫
and π π
π (sin ) (cos )
x
x dx
2
2
4
2
−
∫
are equal, you can
compute the volume by evaluating the integral on the interval 0 4
, π
( ) and doubling the
answer:
2
2
2
2
2
2
2
2
2
0
4
0
4
0
4
π
π
π
π
π
π
π
(cos ) (sin )
cos
sin
s
/
/
/
x
x dx
x dx
x
−
=
=
( )
=
∫
∫
i in
sin
π
π
2
2
0
2
−










=