Unformatted text preview: HW8 Solutions 15.3: Double Integrals in Polar Form
1 0  0  10.
1 2 1y 2 4 x2 + y 2 dxdy = 1 + x2 + y 2 xy 2 dxdy = rdrd = 3/2 /2 0 2 sin 1 4r2 drd = 4 1 + r2 3/2 /2 0 1 1 1 1 + r2 drd = 4  2 14.
0 1(y1)2 /2 1+cos 1 1cos 1 /2 0 0 a /2 0 /2 0 4 sin2 cos r4 drd =  5 8+ 4 18. A = 2
0 3/2 (2 cos + cos2 )d = 26. I0 =
/2 rdrd = 2 + r 4 4 3a2
/2 0 4 29. average = a2 a2  r2 drd = a3 d = 2a 3 15.4: Triple Integrals in Rectangular Coordinates
2 3 0 0 4x2 2 3 0 2 4. = 3 x 2
3 0 2 0 0 1 0 3 2 0 dzdydx = 4  x2 + 4 sin1 0 0 33x 0 1 1x2 0 2 2 3 0 4x2 y 33xy 4x2 x 2 2 0 2 0 4  x2 dydx = 0 3 4  x2 dx = 6 sin1 1 = 3 0 4x2 3 2 4z 2 0 0 3 dzdxdy,
0 4z 2 3 2 0 0 0 4z 2 dydzdx,
0 dydxdz, dxdydz,
0 dxdzdy dzdydx = 3 2
1 1x2 0 0 1 0 10.
0 16.
1 0 0 xdzdydx = x(1  x2  y)dydx 1 x (1  x )  (1  x2 ) dx = 2 1 1 x(1  x2 )2 dx =  (1  x2 )3 2 12 1 =
0 1 12 22.
1 1 0 1 (a)
0  z dydzdx dydxdz dxdydz (b)
0 1 1 1  z (c)
0 0 1  z 1 1 0 1 0  1x2 4x2 0 26. V = 2
0 2 y 0 3x 1 0  1x2 1 dzdydx = 2 dzdydx = 12 ydydx =
0 0 (1  x2 )dx = 2 3 32. V =
2 2  4x2 2 4x x 0 44.
4 0 =
0 sin 2z 4z 2 4x x sin 2z sin 2z dydzdx = dzdx = 4z 4z 0 0 0 4 1 1 1 1 (4  z)dz =  cos 2z =  + sin2 z 2 4 4 2 0 2 4 0 4 0 4z =
0 sin2 4 2 sin 2z 4z xdxdz ...
View
Full Document
 Fall '06
 PANTANO
 Math, Integrals, Multivariable Calculus

Click to edit the document details