- #1

Eclair_de_XII

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- 88

## Homework Statement

"

**Areas of regions**

*Make a sketch of the region and its bounding curves. Find the area of the region.*"

"The region inside the curve ##r = \sqrt{cosθ}## and inside the circle ##r = \frac{\sqrt{2}}{2}##.

## Homework Equations

##A = \frac{1}{2}\int_α^β(f(θ)^2-g(θ)^2)dθ##

*Answer as given by textbook:*##\frac{1}{4}(2-\sqrt{3})+\frac{π}{12}##

## The Attempt at a Solution

Now, since the problem is asking for the region shared by both curves in the first quadrant, the lower limit of my integral would be 0, up until some arbitrary point which I solved for by equating the two curves.

##\sqrt{cosθ} = \frac{\sqrt{2}}{2}##

##cosθ=\frac{1}{2}##

##θ=arccos(\frac{1}{2})=\frac{π}{3}##

So my upper limit would be ##\frac{π}{3}##. Plugging these limits into the equation, I have...

##\frac{1}{2}\int_0^{\frac{π}{3}}(\frac{1}{2}-cosθ)dθ = (\frac{1}{4}θ-\frac{1}{2}sinθ)|_0^{\frac{π}{3}} = \frac{π}{12}-\frac{\sqrt{3}}{4} = \frac{π}{12}-\frac{1}{4}(\sqrt{3})##

So I'm missing like a ##\frac{1}{2}## in my answer, and I don't know why. Could someone tell me what I did wrong?