Area of Polar Curves (NOT a test question)

Question

Find the area enclosed by the cardioid r = 1 − sin θ (4 points)

	3π
	6π
	12π

Thank you in advance.

Answer 1

The cardioid is symmetric about the x axis.

The area formula in polar co-ordinates is 1/2∫r²dθ.

You want the integral from 0 to π of (1-sin θ)² dθ...because of the symmetry.

The integrand becomes cos²θ + 2 sin θ which then becomes

[(1+cos 2θ)/2] + 2 sin θ.

You should be able to integrate this easily and evaluate between the limits.