Question

Find the area of the region which is bounded by the polar curves
theta =pi and
r=2theta 0<theta<1.5pi inclusive

Answer 1

It will help to draw a figure.
I did, and it looks to me like the line theta = pi is NOT a bounding curve. The enclosed area is between the theta = 1.5 line and the spiral cruve r = 2 theta.

For the area, integrate
(1/2) r^2 d theta
= (1/2) 4 theta^2 dtheta
from theta = 0 to theta = 1.5 pi

I get the indefinite interal to be (2/3)(theta)^3

The definite integral is then
(2/3)(3 pi/2)^3

= (9/4) pi^3

Answer 2

The answer is not correct, please try again.

Answer 3

I got it, u have to
integrate (1/2) 4 theta^2 dtheta

from theta=0 to theta=pi