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010.104-2004-2-mid

# 010.104-2004-2-mid - o 2> 5 1r 3r 4 Z 9< s2 j h...

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Unformatted text preview: ; o 2 > 5 2004 10 23{ 1r 3r 4 Z 9 < : s2: j h ]_ \ s&` "r r(8& 100&). HH x h 1. (10&) Find the length of the following curve : h y = log(x + x2 - 1), 1 x 2. 2. 3. (15&) Compute (f (r)- ), where f : R R is a differentiable function, h r - = (x, y, z) in R3 and r = x2 + y 2 + z 2 . r Evaluate the following iterated integrals : 1 1 1 (a) (10&) h 0 y y 1 + x5 dxdy. 1 + y 3 dydx. 1 x (b) (10&) h 0 4. (10&) Find the average mass density of the region h x 0, 1 x2 + y 2 2, where the mass density is given by (x, y) = x2 . 5. 6. (10&) Find the area of the region h 1 x 2, log x y ex . (15&) Let 0 < a < b. If one takes the disc in the xz-plane of radius a h centered at the point (b, 0, 0), and if one rotates the z-axis, one obtains a 3-dimensional solid called solid-torus. Find the volume of this torus. (Hint. use the cylindrical coordinates.) (10&) Compute the triple integral h x2 + R 7. z2 y2 + 4 9 -2 dxdydz, y2 z2 + 9. 4 9 where R is the set of (x, y, z) with 1 x2 + 8. (10&) Let B = {(x, y, z) R3 | x2 +y 2 +z 2 1}. Show that the improper h integral dxdydz 2 + y 2 + z 2 ) B (x converges if and only if < 3 . 2 ...
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