Chapter 10 - Parametric Equations and Polar Coordinates - 10.2 Calculus with Parametric Curves - 10.2 Exercises - Page 695: 35

$A=2{\pi}r^{2}+{\pi}d^{2}$

$x$ = $r\theta-d\sin\theta$ $y$ = $r-d\cos\theta$ $A=\int_a^b y dx$ $=\int_0^{2\pi}(r-d\cos\theta)(r-d\cos\theta)d\theta$ $=\int_0^{2\pi}(r^{2}-2dr\cos\theta+d^{2}\cos^2\theta)d\theta$ $=\left[\theta r^{2}-2dr\sin\theta+\frac{1}{2}d^{2}(\theta+\sin2\theta)\right]_0^{2\pi}$ $=2{\pi}r^{2}+{\pi}d^{2}$