Part II: The Answers
554
Answers
901–1,001
937.
81
40
You can begin by rewriting the expression underneath of the radical to make the
required substitution clearer:
x
x dx
x
x dx
3
2
0
3 2
3
2
2
0
3 2
9 4
3
2
−
=
− ( )
∫
∫
Use the substitution 2
3
x = sinθ to get x = 3
2
sinθ so that dx
d
= 3
2
cosθ θ . You can find
the new limits of integration by using the original limits of integration and the
substitution. If x = 3
2
, then 3
2
3
2
= sinθ , or 1 = sinθ , so that π θ
2
= . Likewise, if x = 0, then
0 3
2
= sinθ so that 0 = sinθ , or 0 = θ . Using this information, you can produce the
following integral:
3
2
9 3
3
2
3
2
9 1
0
2
3
2
4
3
0
2
2
sin
( sin )
cos
sin
s in
co
θ
θ
θ θ
θ
θ
π
π
( ) −
=
( )
−
(
)
∫
∫
d
s s
sin
cos cos
sin cos
θ θ
θ
θ
θ θ
θ
θ θ
π
π
d
d
d
= ( )
=
=
∫
∫
3
2
9
3
2
3
2
4
3
0
2
2
5
4
3
0
2
2
5
4
s sin cos sin
cos
cos sin
2
0
2
2
5
4
2
0
2
2
3
2
1
θ
θ
θ θ
θ
θ
θ θ
π
π
∫
∫
=
−
(
)
d
d
Now use the substitution u = cosθ so that du
d
= − sinθ θ , or − =
du
d
sinθ θ . You can
again find the new limits of integration: If θ π
= 2
, then u =
=
cos π
2
0. Likewise, if θ = 0,
then u = cos 0 = 1. Using the substitution along with the new limits of integration, you
get the following (recall that when you switch the limits of integration, the sign on the
integral changes):
−
−
(
)
=
−
(
)
=
−
(
)
∫
∫
∫
3
2
1
3
2
1
3
2
5
4
2
2
1
0
5
4
2
2
0
1
5
4
2
4
0
1
u u du
u u du
u u du
Now you have the following:
3
2
3
2
3
5
3
2
1
3
1
5
3
2
2
15
5
4
2
4
0
1
5
4
3
5
0
1
5
4
5
4
u u du
u
u
−
(
) =
−






=
−
( )
=
( )
∫
= = 81
40