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Solve the given problems by integration.Find the area under $y=6 \sin ^{2} x \cos x$ from $x=0$ to $x=\pi / 2$.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 1

The General Power Formula

Integrals

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University of Michigan - Ann Arbor

Idaho State University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:54

Evaluate the integrals.

01:24

Evaluate the integral.…

06:01

03:39

Sketch the region of integ…

02:18

Evaluate the following int…

01:16

For the following exercise…

02:20

Find the area of the regio…

03:36

01:45

00:57

Area Find the area of the …

in this question we want to find the area bounded by this curve And x equals to zero to tie over to In general. When we want to find area under a curve with X. From A to B. We will just take a small strip over here. And the wife of this trip is so small it's just a small change in X. Or quite D. X. So over here it has a wide value. Why? So Ariel. This small strip is why the X. Yeah. So to find a total area under the curve right here we're access from A to B. We will just sum up. But because this is a continuous submission it will be an integration sign something from A to B. All the small strips we have so this will become area underground. So back to this expression. The area will be from Mexico 0 to power to Y. G. X. But the one is 69 square X. Course in X the X. Now there are a couple of methods to integrate this. But for this topic we're using substitution method and the integration power rule for the power rule. If I integrate you to the power and respect to you I would want to the power and divide by the new power for indefinite integral. We need to put plus C. For definite integral. We do not need to do so. So let's use substitution. We can next you be equal to sine X. Since there are more sign then co sign here the U. D. X will be called X. So the U. Is co sign X. D. X. So this part is the you. Now we need to substitute the lower and upper limit because these are in X. We want to make them in you because little be using you. So when X equals to zero the lower limit My you is signed zero and they'll be zero. Yeah When X equals two pi over through the upper limit. My you will be signed pie or two. And there is one So we're ready now to substitute. So the zero X equals zero. Now becomes you goes to zero as it goes to power do now becomes you go to one. Now these six can be brought out. Science square X is now since I X is used or sine square eggs will be used square Jose asd excess. Now, d'you Using power rule we will be integrating new square. So using the power rule you square. The power will be added by once every three divided by the new power. Yes. This and this cancer leaving us with two. So we have to you can also stopped the upper limit in and then minus off by something. The lower limit in. So be one cute minus zero. Cute. So the final answer is to you need square

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