area in polar coordinates

Question

Find the area of the region common to the two regions bounded by the following curves.

r=-6cos(tinta)  and r=2-2cos(tinta)

Answer 1

Find an area with curves means we are using integrals. We can either use upper and lower curves or right and left curves to find the area in between. It is important to realize that the first function always needs to be larger than the second.

Our Area equation is Double Integral between r=-6cos(theta) to r=2-2cos(theta) r * dr *dtheta

A= Integral of -6cos(theta) to 0 dtheta times Integral of 0 to 2-2cos(theta) r dr

A = dtheta from -6cos(theta) to zero times Integral of r^2/2 from 0 to 2-2costheta

A = 1/2 integral of (2-2cos(theta))^2 dtheta

A= 1/2 integral (-2(4cos^2theta-8costheta+4) dtheta

A= 6theta - 8 sin(theta) +sin(theta) + constant ---> evaluated at 0 to -6costheta

A=36-8sin(6)+sin(12)-48theta^2sin^4(3)+4theta^4*sin^3(3)(sin(3)+36cos(3)+0 * theta^6

A= 36cos(theta) - 8 sin(6cos(theta)) +sin(12cos(theta))

Check my math but if you are in theta, r, sin and cos then you are in polar

Remember that cos(theta) = x/r and sin (theta) = y/r meaning rcos(theta) = x and y= rsin(theta)