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## Homework Statement

Change the order of the limits of integration of the following double integral and evaluate.

## Homework Equations

[itex] \int_{0}^\frac{\pi}{2} \int_{0}^{cos(\theta)} cos(\theta)\,dr\,d\theta [/itex]

## The Attempt at a Solution

Evaluating as it is, I arrive an answer of [itex] \frac{\pi}{4} [/itex].

I know the region to be integrated is the semicircle bounded by the polar axis, with corner points at r = 0, and r = 1, with a height of 1/2. I know that normally, in the cartesian case, to change the order of integration requires the limits to be written from y(x) to x(y), with the x or y intervals adjust accordingly.

Thus, for this problem, the original region is bounded by:

[itex] 0<r<cos(\theta)[/itex] and [itex] 0<\theta<\frac{\pi}{2} [/itex].

Changing the form, I would write,

[itex] 0<\theta<cos^{-1}(r) [/itex] and [itex] 0<r<1[/itex]

Trying to evaluate in this manner, I end up at

[itex] \int_{0}^1 sin(cos^{-1}(r))\,dr [/itex],

after which I cannot go further. I have very little experience of changing limit orders in the polar case. Any hints would be appreciated!

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