tutorial3 - Answer 1 3 • Integrals and volumes 1 Sketch...

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Tutorial 3 - MAT 1332A - September 28, 2009 Integration by substitution and by parts (at the same time) 1. sin xdx Answer: 2(sin x - x cos x ) + c Integrals and areas 1. Sketch the region bounded by y = e x , bounded below by y = x and on the side by x = 0 and x = 1. Then find the area of this region. Answer: e - 1 . 5 2. Do the same for the region enclosed by the parabolas y = x 2 and y = 2 x - x 2 . Explain how to sketch the graphs of the parabolas and how to find their intersection point.
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Unformatted text preview: Answer: 1 / 3 • Integrals and volumes 1. Sketch the region bounded by the curves y 2 = x and y = x 3 . Revolve this region around the x-axis and find the volume of the resulting solid. Answer: (5 / 14) π 2. Revolve the same region around the y-axis and find the volume. Answer: (2 / 5) π • Partial fractions 1. ∫ 4 x 3-3 x + 5 x 2-2 x Answer: 2 x 2 + 8 x-5 2 ln | x | + 31 2 ln | x-2 | + c...
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