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Numerade Educator

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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

the first thing you know we can do is we know nine X squared plus four y squared equals 36 needs to be written in terms of one. In other words, if it could be written as expert over four plus y squared over nine. Okay, Now, this is an important step because now we know the side length s of the triangle is age over squirt of to this formula's listed in the textbook, which means the area and I saw Sly's right triangle is 1/2 times s squared. So it's gonna be 1/2 times each scored over, too, because when we square this, we can sell off the square root, which gives us a TSH squared over four. Okay, now that we have this, we know we have a chav axe is has to be doubled. So now we have to And then what we found earlier 1/2 times 36 months, nine x squared. As you can probably tell, these two are gonna Kim. So we should end up with squirt of 36 miles, not x squared. It's time to put this in the integral. Now pull out the 1/2 from 0 to 2. 36 minus nine x squared. This is multiplied by D of Axe. We can integrate this now Remember, when we interrupt me, pull out the constant and we can use the power rule, which means increased the expert by one divide by the new exponents. So we end up with this in parentheses and then recall the fact that we have to now substitute in the bounds. So again, one half's on the outside, 72 minutes 24. And that's just minus zero. Between playing with zero, we got zero. This is equivalent to 24 which is our solution.