Homework SourseWhen using the formula 12 integral a to b r outer2 r inner2
When using the formula 1/2 integral a to b (r outer^2 - r inner^2)dtheta....How do I know which curve is the outer and inner. For example 2 cardioids, 1 above x axis and 1 below it??How can I deteermine which is outer and inner?
Solution
1. You solve the two polar equations simultaneously in order to find the limits of integration. You called them a and b but they are actually \'theta radial lines.\' 2. Draw a radial line, emanating from the origin, and between a and b. The first curve your line hits is the inner, the second curve your line hits is the outer. 3. Sometimes you can find the a and b by sketching the two curves and thinking about the theta\'s where the two curves intersect. 4. In polar coordinates, above the x-axis and below the x-axis means nothing. It is always measurement out from the origin (the pole) to the curve r = f(theta).
