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Given that $ \displaystyle \int^{\pi}_0 \sin ^4x \, dx = \frac{3}{8} \pi $, what is $ \displaystyle \int^0_{\pi} \sin ^4\theta \, d\theta $?

$-\frac{3}{8} \pi$

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Missouri State University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

given that the integral of signed to the fourth of X from zero to pi is 3/8 pie. We want to find out the exact same integral, of course, with a different variable with the limits flipped. And this is taking advantage of a property. Yeah, you'll see you over and over again and we'll jump right into it. It's just going to flip the area under the curve to become a negative when you flip the limits of integration. One way to illustrate this is if we looked at the Delta acts, for instance, but this first problem, it would end up being B minus a zero pi minus zero over and And then we could look at the turn, for instance as well. So that would end up really just simplifying too high. I over end now if we did that exact same thing with the next one, and we looked at X here we end up getting zero minus pi over and and then the I ith turned their eventually. But just in terms of the widths themselves, not ignoring even the I ith term. This one is positive. This one is negative. So that is just one demonstration of why needs values end up flipping. And when you flip the limits of integration, then it changes your answer to Ah, negative, whatever it waas.

University of Utah