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Numerade Educator

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Problem 64 Hard Difficulty

Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.

Answer

$\frac{128}{3 \sqrt{3}}$

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Video Transcript

Whereas to solve example nine by taking cross section to be parallel to the line of intersection of the two planes. So an example nine we're told the base of this solid. Well it's bounded by on the top the curve Y equals four X. Squared movies. Look this is the gold. He's looking golden. Really love slowly. We Yeah it's like those little rings. Mm I guess they made a mistake and drawing this. Or it should be. I knew we think about corners long. Weirdly here whose last blonde prison bounded by this circle X squared plus Y squared. Mhm equals 16. Where? Why is greater than or equal to zero? Yeah. Say. Well I found Sarah soon sir. I don't know what. No it looks like you're gray overcomes. Um Washington was red. It was not right. Who's actually why now? Cross sections parallel to the line of intersection are rectangles. It's just just then. Oh yeah I've seen that mural. I told you. I mean it's sad. You know Nick has a fucking blink. I well I imagine a worse person. So we're going to integrate with respect to Y. So we want to solve our equation for X. You have X equals plus or minus the square root of 16 minus Y squared. Yeah. Um Yeah sure Malcolm X. Red phone. Mhm. It's just and therefore we have the width of the base of one rectangle. I'll just call this correct. Um E. Is going to be two times the square root of 16 minus Y squared. Rape the department the height of the rectangle. For a particular why we know will satisfy I call this H. Right Y'all ain't ready to drink that team H over. I'm just permanent. Why Is equal to the tangent of the angle of intersection which is 30°.. And so it follows that our height H. Is equal to Why Times The Tangent of 30°.. And so the cross section area of the rectangle A. Of why? This is of course the base times the height which is two times the square root of 16 minus y squared times Why Times The Tangent of 30°.. You what? This is equal to two of the route 3? Burning about why times the squared of 16 minus y squared. Yeah, mm. And now he is an integral to find the total sum of these cross sections which is our volume. So our volume is the integral from. We see we're y ranges from 0 to 4 of the area A. Of Y. Dy. This is two of the route three times the integral from 0 to 4 of y. Times the square root of 16 minus Y squared. Dy. Now to find this integral, we're going to have to make a U Substitution. Or if you're clever you can do without the U substitution and and do it in your head. We get two of her three times and then we have A negative 1/2 for the negative in front of the Y. Squared and then one half for the square root times and then no sorry a negative one half for the negative Y. Squared and then times a three half for the square root. Yeah but it's the opposite of three halves. It's going to be sorry they reciprocal of three halves which is two thirds times 16 minus Y swear to the new power three halves. This is why I multiplied by two thirds And we're letting wide range from 0 to 4 we sell. So this is if you're thinking without doing the U substitution you just do it in your head plugging in. We eventually get after some steps 128 Over three times the square root of three still talking. And then the juices could taste like, Yeah.