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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.$$y=x, y=0, x=2, x=4 ; \quad \text { about } x=1$$

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$\frac{76}{3} \pi$

04:05

Wen Zheng

02:35

Amrita Bhasin

04:37

Mutahar Mehkri

08:01

Linda Hand

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 2

Volumes

Missouri State University

Oregon State University

Harvey Mudd College

Lectures

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Find the volume of the sol…

02:33

03:58

03:46

08:25

05:36

07:08

06:40

alright. Our goal is to take this region and yellow underneath the line Y equals X. From 2 to 4. And we're gonna rotate it around the line X equals one. Using washers washers in general basically we're gonna do the volume of the disk and then we can subtract out the whole shape disk. So we get pi big R squared. One is little R squared for the net area times. Dy making it a volume. We have to be careful here because if you look carefully at my little radius is that go out from access of rotation to our inner and outer dimensions, we can see that the while big are always goes to the line X equals four. Little are for the bottom half does go to the line X equals two. But for the top half it goes to the line Y equals ax. We have to be super careful here as we set up our integral. Okay so here we go, volume then will be let's do the first the bottom half we'll go from zero to to remember everything has to be in terms of why when we rotate around something vertical. So we are going to now do Big R big are for our bottom half that's just the distance between four and one. So that's just three, three squared. Our little r is just the distance between two and one. So that's one. So that is our first volume integral. Our next one goes in from 2 to 4 and once again our big R is still three squared but r little r now goes from y equals ax to the line X equals one. We're doing everything in terms of why soul write it as Y -1 and then we'll square that. Dy okay so all of this can be worked out. It's not too bad because if we clean it up We just get c. -1. So this is just eight pi but clean that up. So this is just the integral of eight pi. Dy. And the other integral From 2 to 4 is of pie. And then it looks like we have this was a little harder to work on nine minus and then why minus one squared. Um Dy Okay so let's just keep going on this one. The first integral will give me eight. Hi anti derivative is times Y. In assert between zero and two. Um The other one. Second integral. We're going to have nine Y minus Why -1 Cubed Over three. And will integrate. Its really getting squishy there from 2-4. Okay so let's keep going here. So my first uh term that should be a plus. Let's fix that. Okay so the first one I plug into so I'll get a pie plug in two for Y. And then subtract zero. The second term I'm going to plug in four for Y. So I get nine times four minus four minus one. So three cubed over three. That's the first term. Then I've gotta subtract with three with two plugged in. So I get nine times two- and then 2 -1 is one. So 1 Cubed is just one so 1/3. Okay so there's some fraction cleanup to do when you do the fraction cleanup You end up with 76/3 pi so um the area Or the volume by washers a 76 pi over three. And if you're using your calculator to get the integral it's 79.587. Okay. Hopefully that helped have an amazing day.

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