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Unformatted text preview: (a) f ( x ) = x 3 e x 2 ( x 2 +1) 2 (b) f ( x ) = x 2 ± x +1 ( x +1) 3 (c) f ( x ) = 1 x 2 p x 2 ± 36 Problem 4. Compute each of the following de±nite integrals. (a) Find the area under the curve y = sec 2 x between x = 0 and x = ± . Be careful! (b) Find the volume of the solid obtained by rotating the region in the plane bounded by y = x 2 and x = y 2 around the x axis. (c) Show that the surface area of a sphere of radius r is 4 ±r 2 by rotating the circle y = p r 2 ± x 2 around the x axis. 1 2 PAUL SIEGEL, INSTRUCTOR Problem 5. Challenge Problem (will not be graded): Where is the ±aw in the following \proof" that 0 = 1? Let us compute the inde±nite integral R 1 x dx using integration by parts. Set u = 1 =x and dv = dx , so that du = ± 1 =x 2 dx and v = x . We get: Z 1 x dx = uv ± Z v du = x x ± Z x ± 1 x 2 dx = 1 + Z 1 x dx Subtracting R 1 x dx from both sides, we are forced to conclude that 0 = 1!...
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 '08
 WEINERMICHAELDA

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