Let's say I wanted to find the length of the perimeter of a semi-circle using integration with x(t) = cos(t) and y(t) = sin(t). I would do like so:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_{0}^{\pi} \sqrt{sin(t)^2 + cos(t)^2} dt

= \int_{0}^{\pi} \sqrt{1} dt

= \pi[/tex]

now if I wanted to do it for the whole circle I would do it like so:

[tex]\int_{0}^{\pi} \sqrt{sin(2t)^2 + cos(2t)^2} dt

= \int_{0}^{\pi} \sqrt{1} dt

= \pi[/tex]

Now obviously I made a mistake: I didn't replace 2x with u (etc...). But my question is: why can't I do this? What rule stops me from eliminating the cos^2 + sin^2 and getting the wrong answer?

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# Integration and trig

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