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Numerade Educator

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Problem 46 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int^{3\pi/2}_{0} \mid \sin x \mid \,dx $

Answer

$\int_{0}^{3 \pi / 2}|\sin x| d x=3$

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Video Transcript

what makes this problem so tricky is understanding that graph of the absolute value. Because if you were to ask autograph signed the sign curve, I think most people graphic correctly where it goes up at the origin zero, then back down by the time you get to pie. Um And then I'll do this. If it were to continue it would be below the X axis and I'm going to stop it right here because that's three power or two is at the bottom. Um But when you do the absolute value, anything that was below the X axis bounces back above the X axis. Um So what I'm gonna do is I'm actually just gonna do the integral from zero to power to because I know the behavior of the graph of sine of X. Dx. And because I know that this area is going to be the equal to this area which will be equal to this area, which is what we care about. I mean I multiply that answer by three. So to get the correct answer, I mean kind of manipulate the problem this way. So the in um in verse uh anti directive. No, I'm stumbling so much Will be co sign of X. from 0 to Pi or two. Um And it's negative because that's the anti derivative. And then I'm gonna multiply that answer by three. So as you maybe need help thinking about what the unit circle looks like because co sign of power to is zero. I don't actually care about the Y coordinate. And then co sign of zero is one if you want the white cord and say they are. But anyway, the answer will be um zero minus one. Yeah zero minus negative one. Sorry about that. Um So then you end up with three times one, which is three should be your correct answer. And it is.