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Problem 41 Hard Difficulty

Find the volume of the described solid $S .$
The base of $S$ is the same base as in Exercise 40 , but cross-sections perpendicular to the $x$ -axis are squares.

Answer

$$
\frac{1}{3}
$$

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Video Transcript

Okay, so here the on the cross sections are squares, and the lanes of the side of squares are equal to the while to the perpendicular distance from the X access to the line connecting the points 10 and 01 Um, in the x y plane, which is given by this equation here, why is equal to, um, one minus X? So, um, the area of the cross section for the salad is just a is equal to while one minus x square. So on you find the volume just by integrating the cross sectional area as X goes from 0 to 1. So the volume is equal to the integral from 0 to 1 of one minus x quantity squared DX. Okay, well, this is equal to the integral from 0 to 1 square one minus X. Can we get one minus two x plus X squared the X. Okay, so we integrate this while it's not too hard, this is just equal to equal. One is just x, so get X minus on X squared. And then, um, X squared is just 1/3 x cube. So plus 1/3 x cubed. And then, um, we are evaluating this. So, um, 01 So it's plug in one first, and we get one minus one squared plus well, 1/3 times one, uh, cube. And then we are subtracting off. We get we plug in zero. So minus. Well, zero minus while zero squared plus well, 1/3 times zero. That's just a big zero. And we end up with just 1/3. 1/3. So therefore, the volume of the salad is 1/3 cubic units. All right, Thank you.

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