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Let $ V $ be the volume of the solid obtained by rotating about the y-axis the region bounded by $ y = \sqrt{x} $ and y = x^2 $. Find $ V $ both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

$\frac{3 \pi}{10}$

Applications of Integration

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Okay, first finding the region. Well, let's actually sketch the functions. Given the region is this overlapping area. I've shaded it to show now. Cylindrical shells. Well, let's draw our cylinder so we can see what we're looking at. We know this is acts this length. Like this is gonna be squirt of axe minus X. Where now that we have this, you know the height Escort of expense, X squared recall effect. The circumference is two pi X. Therefore, we're gonna be plugging in our bounds from 012 pi axes. I just said times squirt of axe minus X squared times D x. Okay, now that we have this, we know we can pull out our constant to pie. And we know we can clean this up a little bit before we integrate some other words. Simple five. Using the distributive method Now we know we can integrate. Use the power rule, increased the exponents by one divide by the new exponents. X cube becomes extra fourth divide by four. Now that we plug in, we have to part times 2/5 minus 1/4. We just simply plugged in our bounds, which gives us two part times three divided by 20 Which gives us 60 pie six pi over 20 which dividing by two gives US three pi divided by 10 which is our volume.