s9 - University of Toronto at Scarborough Department of...

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H 2007/2008
Solutions #9
1. (a)
integraldisplay
2
- 3
integraldisplay
y
2
0
( x
2
+ y ) dx dy .
integraldisplay
2
- 3
integraldisplay
y
2
0
( x
2
+ y ) dx dy =
integraldisplay
2
- 3
bracketleftbigg
x
3
3
+ yx
bracketrightbigg
y
2
0
dy =
integraldisplay
2
- 3
parenleftbigg
y
6
3
+ y
3
parenrightbigg
dy =
bracketleftbigg
y
7
21
+
y
4
4
bracketrightbigg
2
- 3
=
2 7
21
+
2 4
4
-
parenleftbigg
( - 3) 7
21
+
( - 3) 4
4
parenrightbigg
=
7895
84
.
-3
2
x Equal y
2
-3
2
(b)
integraldisplay
1
- 1
integraldisplay
(2 - y )
2
y 2 / 3
parenleftbigg
3
2
√
x - 2 y
parenrightbigg
dx dy .
integraldisplay
1
- 1
integraldisplay
(2 - y )
2
y 2 / 3
parenleftbigg
3
2
√
x - 2 y
parenrightbigg
dx dy =
integraldisplay
1
- 1
bracketleftbigg
x
3 / 2
- 2 xy
bracketrightbigg
(2 - y )
2
y 2 / 3
dy =
integraldisplay
1
- 1
(
2 -
y
)
3
- 2
(
2 - y
)
2
y -
vextendsingle
vextendsingle
y
vextendsingle
vextendsingle
+ 2 y
5 / 3
dy =
-
bracketleftbigg
(2 - y ) 4
4
bracketrightbigg
1
- 1
- 2
integraldisplay
1
- 1
4 y - 4 y
2
+
1 9
-1
1
x Equal y
2
FractionBarExtFractionBarExtFractionBarExt
3
x Equal LParen1 2 Minus y RParen1
2
1 9
-1
1
y
3
dy - 2
integraldisplay
1
0
y dy + 2
bracketleftbiggparenleftbigg
3
5
parenrightbigg
y
8 / 3
bracketrightbigg
1
- 1
= 20 - 2
bracketleftbigg
2 y
2
-
4
3
y
3
+
1
4
y
4
bracketrightbigg
1
- 1
-
bracketleftbigg
y
2
bracketrightbigg
1
0
+ 0 =
20 - 2
parenleftbigg
-
8
3
parenrightbigg
- 1 =
73
3
.
(c)
integraldisplay
1
- 1
integraldisplay
1
| y |
( x + y )
2
dx dy .
The region
vextendsingle
vextendsingle
y
vextendsingle
vextendsingle
≤ x ≤ 1, - 1 ≤ y ≤ 1 can also be
described as - x ≤ y ≤ x , 0 ≤ x ≤ 1. So changing the
order of integration we have
integraldisplay
1
- 1
integraldisplay
1
| y |
( x + y )
2
dx dy =
integraldisplay
1
0
integraldisplay
x
- x
( x + y )
2
dy dx =
integraldisplay
1
0
bracketleftbigg
( x + y ) 3
3
bracketrightbigg
x
- x
dx =
1
3
integraldisplay
1
0
(
2 x
)
3
- 0 dx =
8
3
integraldisplay
1
0
x 3 dx =
8
3
bracketleftbigg
x 4
4
bracketrightbigg
1
0
=
parenleftbigg
8
3
parenrightbiggparenleftbigg
1
4
parenrightbigg
=
2
3
.
1
-1
1
x Equal VertBar1 y VertBar1
y Equal x
y Equal Minus x
1
-1
1

MATB41H Solutions # 9 page 2
(d)
integraldisplay
π/ 2
0
integraldisplay
cos x
0
y sin x dy dx .
integraldisplay
π/ 2
0
integraldisplay
cos x
0
y sin x dy dx =
integraldisplay
π/ 2
0
sin x
bracketleftbigg
y
2
2
bracketrightbigg
cos x
0
dx =
integraldisplay
π/ 2
0
1
2
cos
2
x sin x dx = -
1
6
cos
3
x
vextendsingle
vextendsingle
vextendsingle
vextendsingle
π/ 2
0
= 0 +
1
6
=
1
6
.
Π
FractionBarExtFractionBarExtFractionBarExtFractionBarExt
2
1
y Equal Cos LBracket1 x RBracket1
Π
FractionBarExtFractionBarExtFractionBarExtFractionBarExt
2
1
(e)
integraldisplay
D
3 y dA, where D is the region bounded by xy 2 = 1, y = x, x = 0 and y = 3.