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Unformatted text preview: = 0, y = 0 and
y 2 = 4 âˆ’ x. So, reversing the order of integration, we get ï¿½ 4 ï¿½ï¿½ âˆš ï¿½ 4âˆ’x x dy
0 ï¿½ 4 dx = 0 âˆš x [y ]0 4âˆ’x dx 0 ï¿½
= 4 âˆš
x 4 âˆ’ x dx 0 ï¿½
ï¿½4 ï¿½ 4
2x
2
3/2
+
(4 âˆ’ x)3/2 dx
= âˆ’ (4 âˆ’ x)
3
03
0
ï¿½
ï¿½4
4
=âˆ’
(4 âˆ’ x)5/2
3Â·5
0
7
2
=
.
3Â·5
2 4. ï¿½ 8 ï¿½ï¿½ âˆš ï¿½ y/3 y dx
0 ï¿½ 12 ï¿½ï¿½ âˆš dy + 0 ï¿½ y/3 âˆš
y âˆ’8 8 y dx 2 ï¿½ ï¿½ï¿½ ï¿½ x2 +8 dy = y dy
2 ï¿½
=
0 2 ï¿½ dx 3x2 0 =
0 ï¿½ y2
2 ï¿½x2 +8
dx
3x2 (x2 + 8)2 (3x2 )2
âˆ’
dx
2
2 8x3
9x5
x5
+
+ 25 x âˆ’
=
2Â·5
3
2Â·5
9 Â· 24
24 26
+
+ 26 âˆ’
=
5
3
5
896
=
.
15
ï¿½ 5. This is a region of type 4; we view this as an elementary region of
type 1. The projection of W onto the xy plane is the elementary region
of type 2 bounded by y = x2 and y = 9.
ï¿½ï¿½ï¿½ ï¿½ 3 ï¿½ï¿½ 9 9âˆ’y ï¿½ï¿½ ï...
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This note was uploaded on 11/23/2013 for the course MATH 18.022 taught by Professor Hartleyrogers during the Fall '06 term at MIT.
 Fall '06
 HartleyRogers
 Calculus

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