a8_soln

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

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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 θ . Solution: To sketch the region we first sketch r = cos 2 θ 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 ≤ θ ≤ π 4 . Thus, we will calculate the area of half of a loop and multiply by 4. This gives A = 4 integraldisplay π/ 4 1 2 (cos 2 θ ) 2 dθ = 2 integraldisplay π/ 4 1 2 (cos 4 θ + 1) dθ = bracketleftbigg 1 4 sin 4 θ + θ bracketrightbigg π/ 4 = π 4 NOTE: To do the problem without using symmetry, we would get that the formula for area is: A = integraltext π/ 4- π/ 4 1 2 (cos 2 θ ) 2 dθ + integraltext 5 π/ 4 3 π/ 4 1 2 (cos 2 θ ) 2 dθ . b) Inside both r = 3 cos θ and outside r = 1 + cos θ ....
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a8_soln - Math 237 Assignment 8 Solutions 1 For each of the...

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