Finding the volume bounded by two equations rotated around y = 2I need to find the volume of a solid generated by a rotating the area bounded by the equations about y = 2:- y = 1- y = x 2 I've graphed the functions, but I'm not sure how to setup an integral to find the volume.

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Finding the volume bounded by two equations rotated around   y  =  2I need to find the volume of a solid generated by a rotating the area bounded by the equations about   y  =  2:-   y  =  1-   y  =      x    2  I&#039;ve graphed the functions, but I&#039;m not sure how to setup an integral to find the volume.

Payten Daniels · Accepted Answer

Step 1Think of this as a washer problem, with the volume resembling a ring. The inner radius is bounded by   y  =  1 while the outer is bounded by   y  =      x    2  . Where they intersect, at       x    2    =  2 or   x  =      2   is your upper bound of integration.Step 2Notice that the inner radius is   2  −  1 because that&#039;s the difference between   y  =  2 and   y  =  1. The outer radius is   2  −      x    2   accordingly. Thus:  π      ∫    0                  2              (  2  −  1      )    2    −            (      2      −              x        2            )        2    d  xDepending on what your lower bound is (it could be   x  =  −      2  ), you can multiply accordingly via symmetry (by 2).

Finding the volume bounded by two equations rotated around y=2: y=1, y=x^2

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