Answer
$D$ = $\frac{\pi}{2}-\frac{8}{3}$
Work Step by Step
$D$ = $\int{\int}xdA$
$D$ = $\int_0^{\frac{\pi}{2}}\int_2^{2\cosθ}r\cosθrdrdθ$
$D$ = $\int_0^{\frac{\pi}{2}}\int_2^{2\cosθ}r^2\cosθdrdθ$
$D$ = $\frac{1}{3}\int_0^{\frac{\pi}{2}}[r^3]_2^{2\cosθ}\cosθdθ$
$D$ = $\frac{8}{3}\int_0^{\frac{\pi}{2}}[\cos^3θ-1]\cosθdθ$
$D$ = $\frac{8}{3}\int_0^{\frac{\pi}{2}}[\cos^4θ-\cosθ]dθ$
$D$ = $\frac{8}{3}\int_0^{\frac{\pi}{2}}[(\frac{1+\cos2θ}{2})^2-\cosθ]dθ$
$D$ = $\frac{2}{3}\int_0^{\frac{\pi}{2}}[1+2\cos2θ+\cos^2{2θ}-\cosθ]dθ$
$D$ = $\frac{2}{3}\int_0^{\frac{\pi}{2}}[1+2\cos2θ+(\frac{1+\cos4θ}{2})-4\cosθ]dθ$
$D$ = $\frac{1}{3}\int_0^{\frac{\pi}{2}}[3+4\cos2θ+\cos4θ-8\cosθ]dθ$
$D$ = $\frac{1}{3}[3θ+2\sin2θ+\frac{1}{4}\sin4θ-8\sinθ]_0^{\frac{\pi}{2}}$
$D$ = $\frac{1}{3}[\frac{3\pi}{2}+2\sin{\pi}+\frac{1}{4}\sin{2\pi}-8\sin{\frac{\pi}{2}}]$
$D$ = $\frac{\pi}{2}-\frac{8}{3}$
