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251sol8

# 251sol8 - fact the right-hand edge of the triangle is the...

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Assignment 8 Multivariate Calculus Math 251 (Fall 2010) A1. A1. Find the average value of the function f ( x ) = integraltext 1 x cos( t 2 ) dt on the interval [0 , 1]. Solution The average is I = 1 1 - 0 integraldisplay 1 0 f ( x ) dx = integraldisplay 1 0 integraldisplay 1 x cos( t 2 ) dtdx. This double integral is over a region which is a triangle in the ( x,t )-plane, with vertices (0 , 0), (0 , 1), and (1 , 1). It is much easier to reverse the order of integration, and using the
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Unformatted text preview: fact the right-hand edge of the triangle is the line x = t . I = I 1 I t cos( t 2 ) dx dt = I 1 b cos( t 2 ) x B t x =0 dt = I 1 t cos( t 2 ) dt. Finally, we substitute u = t 2 so that t dt = 1 2 du . When t = 0, we have u = 0. When t = 1, we have u = 1. I = 1 2 I 1 cos u du = 1 2 (sin 1-sin 0) = sin(1) 2 . a...
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