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## Find the area of the region that lies inside both curves.?

I assume by “inside both curves” you mean their intersection and not their union.

Both circles of radius 3. One centered on y-axis, the other centered on x-axis with (0,0) a common point.

They interesect where 3 cos θ = 3 sin θ, so θ=0 or θ = π/4.

Because of symmetry just use area of onbe of them from 0 to π/4 and double it. So for the sine curve, area =

..π/4

½ ∫ 3² sin² θ dθ

…0

If you double that right now, the area you seek is …

..π/4

∫ 3² sin² θ dθ

.0

This is a basic integral, if you use the formula:

sin²x = (1 – cos 2x)/2

It should be easy to complete now. Just draw the graphs and you’ll get the picture I was describing

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