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Solve the given problems by integration.Find the area bounded by $y=x \sin x,$ the $x$ -axis, (a) between 0 and$\pi,$ (b) between $\pi$ and $2 \pi,(\mathrm{c})$ between $2 \pi$ and $3 \pi .$ Note the pattern.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 7

Integration by Parts

Integrals

Harvey Mudd College

Baylor University

Idaho State University

Lectures

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In mathematics, precalculu…

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Finding Area Find the area…

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Finding a Pattern Find the…

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in this question. Were given this graph, why equals two X in x. We want to find an area. Another ground for X equals zero to pi. As it goes to pi to two pi and as opposed to pine tree pine. Now let area B eight. Now we know that area will be taking why the X integrating it From x equals two A 2 b. So let's find the A real first. We got the lower upper limit. They will be just the king Y dx. So the Y is ex cynics. Yes, we'll be using integration by parts integration by pack. This when we integrate you will respect to be we'll get you ve minus integrate V with respect to you. So let the U B X. So this part here is you and my D V. The sun next E X. So this part over here is my D V. So let's differentiate you. Do you will be just the X. Let's integrate V integrating Sine X. We will get minus cosine X plus C. So let's put everything into the the integration by parts formula. So we'll have UV us X V is minus cosine X minus integrate of V is minus cossacks and you will be the X. So let's see So this 1 -1 because X plus integrate of course science X. I will get say next plus C. So that's my area. So for part A where X is between zero two pie. My area will be yeah I'm just gonna put them into the this will be the epidermis pie locally me is zero so subbing them in. Yeah minus the rest will just be zero. My area will be pie unit square. Yeah for B. Where my ex is pie two x is two pi Okay my area will be after you subbing them in with the upper limit of two Pi minus of the lower limit of pie. Yeah. Yeah. Mhm. Mhm. two Pi minus of the Yeah. Yeah. You will get Trippi unique square for part C. We're x equals two two pi do Trippi mm hmm. The and After after stabbing it and all that. You will get five pi unique square. So we notice a pattern. The pattern is that the answers all multiples of time. So yeah.

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