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Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{4}} \sqrt{1 - \cos 4 \theta} d \theta $

$\frac{\sqrt{2}}{2}$

Integration Techniques

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Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

this problem is from Chapter seven section to problem number forty six in the book Calculus Early Transcendence. ALS eighth edition by James Door Here we have a definite integral of the square root of one minus cosign for theta. So to deal with the square root, let's first deal with this co signing for theta. So coming over here to the right, we can apply the double angle formula and one way to state this is that co sign of two X is one minus two times science where so since we have we don't have a two x web of fourth Ada. We should use X equals to data. So then we have co signs for data is one minus two times science where the two data, then we have one minus cosign for data is simply to sign square to data. And this means that this way room will become square root too. Times square, the science where Susanna So one must be careful here because the square root of something square isn't necessarily the thing that's being swear. It's absolute value. So for that reason to be safe here, we'LL write one too Absolute value signed to theatre. So here, let's make an observation. Due to the bounds of integration, we have that data is between zero before. So this means that tooth data is between zero in power too. So multiply each side of this inequality by two and observed that sign is positive on the interval and separate us Ziona pirates who sinus positive there and sent sinus positive. There's no need for the absolute value sign so we could simply right I went to times scientific data, Islamics the thing that we're integrating much easier to work with. So coming back to the original problem we have in general sealed up our floor. I grew into stone scientific data separate our work on the side Also here in this new inaugural and my help you to go ahead and do it yourself here you could take you to be to theatre, not necessary. But it might help. So here we have negative won't too Coastline of two data. It's over too. And we have the end points. See our own power before, so it's gotta unplug those in. So it's plot this constant first, So plugging in pi over four, we have co sign a power too minus cause I know. Zero. So we have got you noting from the unit circle. This is zero. That's the one so determined. The parentheses is minus one and we have another minus one out here. So what you got? And cancel out those negatives. Take it square, too over to, and that's our answer.