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Problem 55 Hard Difficulty

Find the volume of the described solid $ S $.
The base of $ S $ is an elliptical region with boundary curve $ 9x^2 + 4y^2 = 36 $. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Answer

24

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Video Transcript

would love to just were given a solid S and rest to find the volume of this solid. We're told the base of S. Is an elliptical region with a boundary curve. Nine X squared plus four Y squared equals 36. Thank you. And a cross sections perpendicular to the X axis are isosceles right triangles with a high pot in use in the base course It's got 100 clear and trade That one off the board. So the hypothesis of an isosceles right triangle. Right right. I'll call this each. We know the H satisfies S. Squared plus S. Squared. Since isosceles equals H. Squared. Where S. Are the other sides of the isosceles right triangle. Therefore we have that S. Is equal to H. Over the square root of two. And the area of our societies triangle is one half times the base which is age times the height. Uh huh. Or okay one half times S squared. Which in terms of H is one half times H squared over two which is H squared over four. Mcdonald's guilty or not guilty. Now to find a church, let's look back at our lips. So the top half of the ellipse you can find by taking the positive square root with respect, right? Why is a function of X? So we have that just lie. Why is equal to the positive square root of 1/4 of 36 -9 x squared. Which is the same as one half times the square root of 36 minus nine X squared. This is the top and the bottom of the ellipse is given by the equation Y equals negative one half times the square root of 36 minus nine X squared. Yes. Now either by symmetry or by using top minus bottom, we have the H as a function of X. Is going to be two times one half times the square root of 36 minus nine X squared. Which is just the same as the square root of 36 minus nine X squared. It's got to answer. And therefore the volume of the solid form from the cross section interpretation is the integral of the some of the areas of each triangle. It's the volume. V is the integral from. Well, we're going to let X range from, well, if you look at our lips, X range is from negative to to positive too. Of the area A of X. Dx. Now our area A of X. Well this is H of X squared over four. Yes. Without any practice song. I'll be and plugging in. This is the integral from negative 2 to 2 of 36 minus X squared over four dx drummed in. Yeah, that's that's always very that's awesome. You guys, and This is an easy integral to evaluate. Once you do, you should find this is 24