Finding volume using disks/washersFind the volume of a solid obtained by rotating the region bounded by y = x and y = x about y = 2.

Question

Finding volume using disks/washersFind the volume of a solid obtained by rotating the region bounded by   y  =      x   and   y  =  x about   y  =  2.

Spielgutq1 · Accepted Answer

Step 1Here, I am going to use the washer method because that is often easier when there are more than two curves.First, graph on a graphing calculator or visualize the graphs of   y  =      x   and   y  =  x. It is easy to see that   y  =      x   is closer to   y  =  2 than   y  =  x. Also, the region between   y  =      x   and   y  =  x starts at   x  =  0 and ends at   x  =  1. Therefore, since   y  =      x   is closer to   y  =  2, the inside function is   2  −      x   (or       x    −  2; it does not matter since we are going to square this function, which will lead to the same result both ways). Also, since   y  =  x is farther from   y  =  2, the outside function is   2  −  x.Step 2Thus, we have to integrate   [      R    O    (  x  )      ]    2    −  [      R    I    (  x  )      ]    2   (      R    O   is the outside function and       R    I   is the inside function) from the beginning of the region, 0, to the end of the region, 1, and then multiply that by   π, which gives us:  π      ∫    0    1    {  [  2  −  x      ]    2    −  [  2  −      x        ]    2    }  d  xThus, you need to solve that definite integral.

Find the volume of a solid obtained by rotating the region bounded by y=sqrt{x} and y=x about y=2.

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