Calculus Help Please
Calculus

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Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
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Drawing a picture is always a good thing to do for these types of problems.
Start by sketching the graphs of y=8, y=x3, and x=0. We then see that the region to be revolved about the xaxis is in the first quadrant. It is bounded above by the line y=8, on the left by the yaxis, and on the right by the graph of y=x3. Note that the point of intersection in the upper right is (2,8) and can be found by solving the equation 8=x3.
The region is shown, shaded in blue, below:
Now, you generate the solid by revolving this region about the xaxis. The cylindrical shells are generated by revolving a horizontal line segment, shown in green in the diagram, at a fixed yvalue about the xaxis.
These line segments "start" at y=0 and "end" at y=8. Thus the integral giving the volume of the solid of revolution is with respect to y and is of the form
∫y=8y=02πry⋅hydy
where ry is the radius of the shell at y and hy is the height of the shell at y.The height of the shell at y is the length of the line segment at y. Since the length of the line segment at y is the xcoordinate of its right hand endpoint, we have
hy=y1/3.
Keep in mind that we want to write hy in terms of y, since the integral is with respect to y.
The radius of the shell is the height above the xaxis of the line segment:
ry=y.
Thus, the volume of the solid of revolution is:
∫802πy⋅y1/3dy=∫802π⋅y4/3dy=6π7y7/3∣∣80=6π7⋅87/3=6π7⋅27=6π7⋅128=768π7.
Please let me know if you need any clarification. I'm always happy to answer your questions.
is it 768pi/7?
Wait where did the 8 even come from? This isn't my problem?
Yes bro
You literally didn't even answer my own problem, please stop answering my questions, I don't need your help.
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