PHY 206 Midterm: PHY 206 UMiami 160S11 Exam 1
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Fall 2006 x y sin x dx dy. Since the integral r sin x x dx is too hard, we change the order of integration so that we integrate with respect to y rst. This double integral is taken over a region. R, which is de ned by the inequalities 0 y 1 and y x 1. Graphing this region, it is clear that it is the triangle with vertices (0, 0), (1, 0), (1, 1). Thus it is also de ned by the inequalities 0 x 1 and 0 y x. 0 z 1 sin x dx dy dx sin x sin x x y. 1 + x2 + y2 dx dy where r is the region bounded by the top half of the. Convert to polar coordinates by making the substitutions x = r cos and y = r sin . r is then described by the inequalities 0 r 1 and 0 .
