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Problem 43 Hard Difficulty

Find the volume of the described solid $S .$
The base of $S$ is the same base as in Exercise $42,$ but cross-sections perpendicular to the $x$ -axis are isosceles triangles
with height equal to the base.

Answer

$$
\frac{8}{15} \text { units }^{3}
$$

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Video Transcript

So here the cross sections are triangles. So therefore the area is found where all by taking to the area is gonna 1/2 base times height. But in this case, the the base and the height are equal. So therefore we have we have that, um, the area right is equal to just 1/2 be square. Okay, um, and the base is clearly the height of the parabola of a given value of X. In other words, the base length is equal to the Y value. So we get that the area of the cross section of this salad. The area is equal to 1/2 times. Well, one minus X squared square. Okay, now we get the volume here by integrating the cross sectional area. Um, as so, we basically square this thing out and we get that the abuses, um, to the volume right is equal to well, the integral from well, from 01 of we end up getting here one minus two X square, Um, plus X to the fourth, the X. Okay. And now he's go ahead and integrate this term by term. So the volume well is equal Chew, We just get um, they're gonna want it. Just acts and then minus two x squared. So we add one to the exponents. So three and then divide by that three. So we get minus 2/3 X cube, and then we're integrating exit 1/4. So we get a well plus 1/5 X to the fifth, and then we are evaluating it from 01 Okay, so what we get is, well, one minus, um, 2/3 times one. Ah, Cube Plus Well, 1/5 times. One to the fifth. Okay. And then well, what? We're subtracting off. We get, we plug in zero. But that's just, well, zero minus 2/3 times zero, uh, I guess cube. Right? And then plus 1/5 times, zero to the fifth. Okay, but this whole thing is gonna be zero, right? So we just end up with, um Well, one minus 2/3 plus 1/5 and that is equal to well, eat 15th 8/15. So therefore, the volume of the solid, um, our volume off the salad is equal to Hey, Whoops. 8 15 um, cubic units. All right, take care.