I'm trying to calculate the area of the region bounded by one loop of the graph for the equation
$$
r = 1 + 3 \sin \theta
$$
I first plot the graph as a limaçon with a maximum outer loop at $(4, \frac{\pi}{2})$ and a minimum inner loop at $(-2, -\frac{3 \pi}{2})$. I then note the graph is symmetric with respect to the $\frac{\pi}{2}$ axis and the zero for the right half is at $\theta = \arcsin(-\frac{1}{3})$.
So, I chose the interval $[\arcsin(-\frac{1}{3}),\frac{\pi}{2}]$ to calculate the area which can then be multiplied by $2$ for the other half. The problem is that the answer in the book seems to use $\arcsin(\frac{1}{3})$ instead, note the change of sign.
Just to make sure I'm not misunderstanding where I went wrong, I get the answer
$$
\frac{11 \pi}{4} – \frac{11}{2} \arcsin(-\frac{1}{3}) + 3 \sqrt 2
$$
Whereas the book gets
$$
\frac{11 \pi}{4} – \frac{11}{2} \arcsin(\frac{1}{3}) – 3 \sqrt 2
$$
It's a subtle change of sign but I'd really like to understand where I went wrong.
Best Answer
Notice how $\arcsin(-\frac{1}{3}) = - \arcsin(\frac{1}{3})$, so your answer now looks like
$$ \frac{11 \pi}{4} + \frac{11}{2} \arcsin(\frac{1}{3}) + 3 \sqrt 2 \\ $$
That means your area is greater than the answer in your book by:
$$ 2 \left(\frac{11}{2} \arcsin(\frac{1}{3}) + 3 \sqrt 2\right) $$
This might indicate you are calculating the area of the outer loop whereas your book is calculating the inner loop. If you choose the interval $[\frac{3 \pi}{2}, 2 \pi - \arcsin(\frac{1}{3})]$ to calculate the half as you did before, you get:
$$ \begin{eqnarray} A &=& 2 \times \frac{1}{2} \int_{\frac{3 \pi}{2}}^{2 \pi - \arcsin \frac{1}{3}} (1 + 3 \sin \theta)^2 \, \textrm{d}\theta \\ &=& \left[\frac{11 \theta}{2} - 6 \cos \theta - \frac{9 \sin(2 \theta)}{4} \right]_{\frac{3 \pi}{2}}^{2 \pi - \arcsin \frac{1}{3}} \\ &=& \frac{11 \pi}{4} - \frac{11}{2} \arcsin(\frac{1}{3}) - 3 \sqrt 2 \\ \end{eqnarray} $$
This seems to agree with the answer in your book.