Need help with this question:
Find the area of the butterfly region $\int\int 1 dA $, where $R$ is given in polar coordinates as$ 0 ≤ r ≤ r(θ)$ where $r(θ)$ is defined as $r(θ) = 8−sin(θ) + 2 sin(3θ) + 2 sin(5θ)−sin(7θ) + 3 cos(2θ)−2 cos(4θ)$
So I calculated the two integrals using the following bounds: $\int_0^{2π}$$\int_0^{r(θ)} rdrd(θ).$ However, this doesn't necessarily work because you get a value of 0 after solving. Thus, should I have taken the first quadrant and multiplied by 4? PLEASE help!!!!!!
Noting $$ \int_0^{2\pi}\sin(2m\pi)\sin(n\pi)dx=0, \text{ if }m\neq n, \int_0^{2\pi}\sin(2m\pi)\cos(n\pi)dx=0, \int_0^{2\pi}\sin^2(n\pi)dx=\int_0^{2\pi}\cos^2(n\pi)dx=\pi, $$ one has \begin{eqnarray} A&=&\frac12\int_0^{2\pi}r^2(θ)d\theta\\ &=&\frac12\int_0^{2\pi}(8−\sin(θ) + 2 \sin(3θ) + 2 \sin(5θ)−\sin(7θ) + 3 \cos(2θ)−2 \cos(4θ))^2d\theta\\ &=&\frac12(128\pi+\pi+4\pi+4\pi+\pi+9\pi+4\pi)\\ &=&\frac{151}{2}. \end{eqnarray}