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Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = \sqrt[3]{x} $ , $ y = 0 $ , $ x = 1 $

$=\frac{6}{7} \pi$

Applications of Integration

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first thing we should do is we should draw the function just a rough sketch. And then we know that if this is our so called shaded region, then we can consider perhaps that we have a cylindrical shell that looks a little something like this. As you can see, this is the 0.1 comer one as specified in the problem. Now, remember, the formula for volume is from A to B two pi r h d x. Remember, ours acts h is ex the 1/3 or the cube root of X. Now that we've given this, we can pull into the volume formula what we have, which means we can we can pull out the constant which is to pie. We have a bound instead of a to B. We have 01 We have acts, which is our times X to the 1/3 of the cube root of ax exponents are easier to manage because now we're gonna be using the power rule, which means we're gonna be increasing the exponents by one and then dividing by the new exponents, which means we now have six pi over 73 times to a six. As you can see, we're divided by seven times one minus zero, which gives us six pi divided by seven