liggett_midterm1solutions

liggett_midterm1solutions - T. Liggett Mathematics 31B/4...

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T. Liggett Mathematics 31B/4 October 22, 2004 Solutions 1 (15) 1. (a) Find the average value of the function f ( x ) = x 2 1 + x 3 on the interval [0,2]. Average value = 1 2 Z 2 0 x 2 p 1 + x 3 dx = 1 6 Z 9 1 udu = 26 9 . (b) Give a precise statement of the Mean Value Theorem for Integrals. See page 403 of the text. (c) If g is continuous and R 3 1 g ( x ) dx = 8, explain why g must take the value 4 at least once in the interval [1,3]. By the Mean Value Theorem for Integrals, there is a c in [1 , 3] so that 8 = R 3 1 g ( x ) dx = 2 g ( c ) . For this c , g ( c ) = 4. (15) 2. Find the number a so that the line x = a bisects the area under the curve y = 1 /x 2 , 1 x 4. The area corresponding to 1 x a is Z a 1 1 /x 2 dx = a - 1 a . Therefore the a we want is the solution of ( a - 1) /a = (1 / 2)(3 / 4) , or a = 8 / 5. (15) 3. Use slices to find the volume of the solid obtained by rotating the region bounded by y = x and y = x about the line x = 2. Volume =
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liggett_midterm1solutions - T. Liggett Mathematics 31B/4...

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