This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Polar Coordinates

Click to edit the document details