Part II: The Answers
390
Answers
601–700
641.
125
6
Notice that you can easily solve the two equations for x, so integrating with respect to
y makes sense. If you were to solve the first equation for y in order to integrate with
respect to x, you’d have to use more than one integral to set up the area.
Begin by solving the second equation for x to get x = y + 7. Then find the points of intersection by setting the expressions equal to each other and solving for y:
y
y
y
y
y
y
y
+ = +
=
− −
= −
+
= −
7 1
0
6
0
3
2
3 2
2
2
(
)(
)
,
To determine which curve has the larger x values for y on the interval (–2, 3), pick a
point in the interval and substitute it into each equation. If y = 0, then x = 1 + 0
2
= 1 and
x = 0 + 7 = 7. Therefore, the integral to find the area is
(
)
y
y
dy
y
y dy
y
y
y
+ − +
(
)
=
+ −
=
+
−
−
−
−
∫
∫
7
1
6
2
6
3
2
2
3
2
2
3
2
3
2
3 3
2
3
3
2
3
2
6 3
3
3
2
2
6 2
2
3
125
6
=
+
−
−
−
+ − −
−
=
( )
( )
( )
( )
( )
( )
The following figure shows the region bounded by the given curves:
390
Answers
601–700
641.
125
6
Notice that you can easily solve the two equations for x, so integrating with respect to
y makes sense. If you were to solve the first equation for y in order to integrate with
respect to x, you’d have to use more than one integral to set up the area.
Begin by solving the second equation for x to get x = y + 7. Then find the points of intersection by setting the expressions equal to each other and solving for y:
y
y
y
y
y
y
y
+ = +
=
− −
= −
+
= −
7 1
0
6
0
3
2
3 2
2
2
(
)(
)
,
To determine which curve has the larger x values for y on the interval (–2, 3), pick a
point in the interval and substitute it into each equation. If y = 0, then x = 1 + 0
2
= 1 and
x = 0 + 7 = 7. Therefore, the integral to find the area is
(
)
y
y
dy
y
y dy
y
y
y
+ − +
(
)
=
+ −
=
+
−
−
−
−
∫
∫
7
1
6
2
6
3
2
2
3
2
2
3
2
3
2
3 3
2
3
3
2
3
2
6 3
3
3
2
2
6 2
2
3
125
6
=
+
−
−
−
+ − −
−
=
( )
( )
( )
( )
( )
( )
The following figure shows the region bounded by the given curves: