Find the area of the region that lies inside both curves. r = 5 sin(θ), r = 5 cos(θ)

Question

I am completely blanking on this question and I really don't even know where to start.

Answer 1

1. What do the curves look like? Sketch!
2. Where do they intersect?
3. What do you know about area in polar coordinates?

Answer 2

Solving for the point of intersection of the two curves, we get $$5\sin \theta =5\cos \theta $$ $$\Rightarrow \tan \theta =1$$ $$\Rightarrow \theta =\frac {\pi}{4} \mid \frac {5\pi}{4} $$ where $\mid $ stands for "or".

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To calculate the area, we use the formula $$\frac {1}{2}\int_{a}^{b}(r_1^2-r_2^2) d\theta $$ where $a $ and $b$ are the coordinates of intersection.

Now using $r_1 = 5\sin \theta $ and $r_2 = 5\cos \theta $ and $a =\frac {\pi}{4} $ and $B =\frac {5\pi}{4} $, you shall be able to easily calculate your integral.