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Do we absolutely have to change the limits of integration for this problem?

from $\int_{0}^{4\pi}$ to $\int_{0}^{\pi}$?

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You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". Here's a basic tutorial and quick reference. There's an "edit" link under the question. – joriki Oct 13 '12 at 16:05
$\pi$ is a Greek letter, spelled "pi" and you can use it in your post by writing \pi enlclosed in single dollar signs. – chris Oct 13 '12 at 16:06
up vote 0 down vote accepted

If you graph $t\mapsto |\sin t|\sqrt{\cos^2t+1}$ you will find that the part of it between $0$ and $2\pi$ consist of four identical pieces, though two of them are reversed. The reversals do not change the area under each pieces, so you can find the entire integral by integrating just one of them and then multiplying the result by $4$.

(This is progress partly because it allows you to get rid of the absolute value by restricting your attention to a range where $\sin t$ is non-negative).

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