How can I find the volume of a solid of revolutionFind the volume of the solid obtained by rotating the region bounded by y = x 2 and x = y 2 . Rotating about y = 1.I got an intercept of those functions which was (1,1). I tried to use washer method then I got π ∫ 0 1 [ ( x ) 2 − ( x ) 2 ] d x and I took integral of the functions but my volume was not right number. I think my way to solve was not right. Could you post correct way to find the volume?

Question

How can I find the volume of a solid of revolutionFind the volume of the solid obtained by rotating the region bounded by   y  =      x          2       and   x  =      y          2      . Rotating about   y  =  1.I got an intercept of those functions which was (1,1). I tried to use washer method then I got  π      ∫    0    1    [  (  x      )    2    −  (      x        )    2    ]  d  x and I took integral of the functions but my volume was not right number. I think my way to solve was not right. Could you post correct way to find the volume?

Evelin Castillo · Accepted Answer

Step 1Your formula assumes that you are rotating about the x-axis (that is, about   y  =  0), but you aren&#039;t; you are rotating about   y  =  1. Also, you forgot that your other function was   y  =      x    2   and not   y  =  x(the   (  x      )    2   factor was probably meant to be   (      x    2        )    2  ?).If you have a thin strip of width   Δ  x that starts at the point       x    i  , then you have a washer with outer radius   1  −      x    i    2   and inner radius   1  −            x      i       (draw a picture, and remember the axis of rotation). So the volume described by the washer is approximated by   π      (    (    1    −          x      i      2              )      2        −    (    1    −                  x        i                    )      2        )    Δ  x  .Step 2Take the Riemann sum, then limits as   Δ  x  →  0 to turn it into an integral.

Find the volume of the solid obtained by rotating the region bounded by y=x^2 and x=y^2. Rotating about y=1.

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