Unformatted text preview: ¿½ ï¿½ 8xyz dx dy dz =
W 8xyz dz dy dx
ï¿½ï¿½
ï¿½ 3 ï¿½ï¿½ 9 ï¿½ï¿½ 9âˆ’y
=8
x
y
z dz dy dx
âˆ’3
x2
0
ï¿½ 3 ï¿½ï¿½ 9 ï¿½ 2 ï¿½9âˆ’y ï¿½
z
=8
x
y
dy dx
20
âˆ’3
x2
ï¿½
ï¿½ 3 ï¿½ï¿½ 9
y (9 âˆ’ y )2
=8
x
dy dx
2
âˆ’3
x2
ï¿½
ï¿½ 3 ï¿½ï¿½ 9
2
3
=4
x
81y âˆ’ 18y + y dy dx
âˆ’3 âˆ’3
3 x2 0 x2 ï¿½9
81y 2
y4
3
=4
x
âˆ’ 6y +
dx
2
4 x2
âˆ’3
ï¿½
ï¿½ 3ï¿½ 8
3
38
81x3
x9
7
âˆ’2Â·3 +
=4
xâˆ’
+ 6x7 âˆ’ dx
2
4
2
4
âˆ’3
= 0,
ï¿½ ï¿½ 3 ï¿½2
0 as x, x3 , x7 and x9 are all odd functions. In retrospect, we could have
decide very early on that the integral is zero; 9 ï¿½
J (x) = 9âˆ’y ï¿½ï¿½
y ï¿½
z dz dy, x2 0 is clearly an even function of x, so that...
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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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