a8_soln - Math 237 Assignment 8 Solutions 1. For each of...

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Unformatted text preview: Math 237 Assignment 8 Solutions 1. For each of the indicated regions in polar coordinates, sketch the region and find the area. a) The region enclosed by r = cos 2 . Solution: To sketch the region we first sketch r = cos 2 in Cartesian coordinates and then use this to plot the graph in polar coordinates. This gives: From drawing the picture we see that we got half of a loop using 0 4 . Thus, we will calculate the area of half of a loop and multiply by 4. This gives A = 4 integraldisplay / 4 1 2 (cos 2 ) 2 d = 2 integraldisplay / 4 1 2 (cos 4 + 1) d = bracketleftbigg 1 4 sin 4 + bracketrightbigg / 4 = 4 NOTE: To do the problem without using symmetry, we would get that the formula for area is: A = integraltext / 4- / 4 1 2 (cos 2 ) 2 d + integraltext 5 / 4 3 / 4 1 2 (cos 2 ) 2 d . b) Inside both r = 3 cos and outside r = 1 + cos ....
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a8_soln - Math 237 Assignment 8 Solutions 1. For each of...

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