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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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1 Answer 1

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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