How do you find the volume bounded by y 2 = x 3 and y = x 2 revolved about the y-axis?

Question

How do you find the volume bounded by       y          2        =      x          3       and   y  =      x          2       revolved about the y-axis?

lilao8x · Accepted Answer

We need to find where these two curves intersect to find the bounds of integration.      y          2        =      x          3       and   y  =      x          2      , squaring the second expression,       y          2        =      x          4       Solving for       y          2      ,...  [      x          4        =      x          3        ]  i  .  e  ,      x          3        [  x  −  1  ]  =  0.So x=1, x=0 are the points of intersection.From the graphs of these expressions in can be seen that   y  =            x              3             has a greater area than   y  =      x          2       so we must find the area under   y  =      x          2       and subtract it from the area under   y  =            x              3             and then revolve this area about the x axis between the bounds x=1, x=0Volume of revolution is given by   π      ∫          a              b            y          2        d  x So the volume of revolution =  [  π      ∫          0              1            x          3        d  x  −  π      ∫          0              1            x          4        d  x  ]  =  π  [            x              4              4    −            x              5              5    ] [after integration by the general power rule] and evaluated for x=1, x=0 will result in   π  [      1    4    −      1    5    ]  =      π    20  .

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