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Unformatted text preview: Solution to Set 8
1 1 0 2 1. (a)
0 ex+y dxdy =
4−y 2 1 0 2 ex dx x2 2 √
0 1 0 4−y 2 ey dy 1 2 = (e − 1)2
2 0 √
0 (b)
0 xdxdy = 0 dy = (4 − y 2 )dy = 8 3 2. The reversed interated integral is
1 x 0 (a)
0 ex dydx = √ x
0 2 1 0 ex xdx = 2 1 x2 e 2 1 =
0 1 (e − 1) 2 π 2/3 2 4 − cos x3/2 = 3 3 0
y /2 0 4 π 2/3 (b)
0 sin x 3/2 π 2/3 dydx =
y /2 0 0 (sin x3/2 )x1/2 dx =
4 4 3. (a)
R (x + y )dA =
0 0 √ 0 (x + y )dxdy =
0 0 x2 + yx 2 dy = 0 52 40 y dy = 8 3 16−x2 (b)
R xdA = −4 xdydx = −4 x 16 − x2 dx = − 64 3
5 2π 5 0 4. (a)
R xydA = 0 π 0 r2 cos θ sin θ rdrdθ = 2π 0 π 0 cos θ sin θdθ 0 r3 dr =0 (b)
R x2 dA = 4 sin θ 0 r2 cos2 θ rdrdθ = 43 cos2 θ sin4 θdθ = 4π 5. (a) The projection R of the portion of the plane on the xyplane is the triangle bounded by the two axes and the line x = 8 − 2y . 1 A unit normal to the surface is n = √ (i + 2j + 3k). 14 3 n·k = √ 14 √ √ 4 8−2y 16 14 dA 14 dxdy = . Surface area = = 3 3 n · k  00
R (b) The intersection of the paraboloid with the xyplane is the circle x2 + y 2 = 9. The projection R of the portion of paraboloid on the xyplane is the disk with radius 3. 1 A unit normal vector to the paraboloid is n = (−2xi − 2y j − k ). 2 + 4y 2 + 1 4x dA 4x2 + 4y 2 + 1 dA. Surface area = = n · k 
R In terms of polar coordinates, the integral becomes 2π 3 π 4r2 + 1 rdrdθ = (373/2 − 1) ≈ 37.3 π . 6 0 0 1 6. (a) R is the annular region between x2 + y 2 = 1 and x2 + y 2 = 4; z 2 dS =
σ R (x2 + y 2 ) √ x2 x2 y2 +2 + 1 dA 2 +y x + y2
2π 2 1 = √ 2
R (x2 + y 2 )dA = 2
0 r3 drdθ = 15 √ π 2. 2 √ (b) The projection of the surface σ (parameterized by z = + 1 − x2 ) on the xy plane is the region R = [−1, 1] × [0, 1]. The suface integral becomes x2 ydS =
σ 1 0 1 −1 x2 y −x √ 1 − x2 2 + 1 dxdy = 1 2 1 −1 x2 √ dx. 1 − x2
π /2 With the substitution u = sin x, the integral can be evaluated as Therefore,
σ −π/2 sin2 u du = π/2. x y dS = π/4. 2 2 ...
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This note was uploaded on 02/27/2010 for the course MATH MATH101 taught by Professor Chan during the Fall '09 term at HKUST.
 Fall '09
 CHAN
 Multivariable Calculus

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