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Evaluate the integrals.$$\int_{0}^{2} \sqrt{4-x^{2}} d x$$

$\int_{0}^{2} \sqrt{4-x^{2}} d x=\pi$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

Integration Techniques

Section 3

Trigonometric Techniques of Integration

Integrals

Integration

Campbell University

Harvey Mudd College

Baylor University

Lectures

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So since we have a square root of this insane term, we'll be using yourself. Well, this is of the form acceptable to four fine vato. So let's, um, rewrite this as X over four is equal to sign a bit. And I'm looking drop our triangle in case we need it. So opposite over I Pont news, the remaining it is four square that 16 minus x squared. Okay, So what do you write this? So we have the square roots of four minus x X is four squared. Actually, that should be two squared, right? Yeah. So this election for and this is for as well, yeah. So yet four sine squared. And then Well, that's fine. DX, that's equal to to co sign up there. Yeah. So yet we contract out of four. The square root afford to do we get to square roots of one money signing squared of data. So that's coastline square. DaVita my art degree in identity. And then we have DX. That's, um, to so that's four whole sign of data. You data. Okay, so taking the square root, I get coastline. We stopped two of them to get four co sign squared of data. You data. Okay, so I'll use my half uncle identity so I can write this us for one plus co sign too. Data over to the data that gives me to in a girl of one plus co sign of two data Ubeda and I'm taking our interval. We get to see the times they don't plus sign of two data over to. And now we want to about weight that are at our endpoint. So let's actually change our endpoint. So we have when actually good to we have sought to sign up to And then what x is equal to zero we have, actually, we're solving for theater. So when exiting to we get to sign, they don't. So that's one is equal to sign of data. So data is able to sign inverse of one. And then we also get better is equal to, um let's see what x zero, we get signed interests. They're all so we have sine inverse of one on sign, Injurious of zero. Okay, so yet, to our data is sine inverse of one plus fine of two sine inverse of one over two. And then we get minus two times fine. Inverse of zero plus sign of two sine inverse. Observe over two. Okay, so we get before execution.

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