Part II: The Answers
402
Answers
601–700
657.
3
2
Begin by finding all the points of intersection of the three functions. To do so, solve
each equation for y and then set the functions equal to each other. Solving the second
equation for y gives you y
x
= − 2
, and solving the third equation for y gives you
y = 3 – 2x. Set these two functions equal to each other and solve for x:
− = −
=
=
x
x
x
x
2
3 2
3
2
3
2
Likewise, finding the other points of intersection gives you x
x
= − 2
so that x = 0 is a
solution and gives you x = 3 – 2x so that x = 1 is a solution.
Notice that on the interval (0, 1), the region is bounded above by the function y = x and
below by the function y
x
= − 2
. On the interval (1, 2), the region is bounded above by
the function y = 3 – 2x and below by the function y
x
= − 2
. Therefore, the integrals to
find the area of the region are
x
x dx
x
x dx
x dx
x
− −
( )





 +
−
(
)− −
( )

 

 
=
+
−
(
∫
∫
∫
2
3 2
2
3
2
3 3
2
0
1
1
2
0
1
) )
= ( ) + −
(
)
= +
−
(
)− −
( )






=
∫ 1
2
2
0
1
2
1
2
3
4
3
3
4
3
4
6 3
3 3
4
3
2
dx
x
x
x
658.
4
3
Begin by finding the points of intersection by setting the functions equal to each other.
Square both sides of the equation and factor to solve for x:
x
x
x
x
x
x
x
x
x
x
x x
x
+ = +
+ =
+
+
+ =
+
+
=
+
=
+
= −
4
4
2
4
8
16
4
4
16
8
16
0
4
0
4
0 4
2
2
2
(
)
,