Revolving a function around y-axis, finding volume using shell methodThe region bounded by y = x x 3 + 8 , the x-axis, and the line x = 2 is revolved about the y-axis. Find the volume of the solid generated this way.

Question

Revolving a function around y-axis, finding volume using shell methodThe region bounded by   y  =      x                  x        3            +      8      , the x-axis, and the line   x  =  2 is revolved about the y-axis. Find the volume of the solid generated this way.

Payton Mcbride · Accepted Answer

Step 1For the rotation around y axis the formula is  V  =  2  π      ∫          a              b        x  f  (  x  )    d  xIn this case we have   V  =  2  π      ∫    0    2              x      2                      x        3            +      8          d  x  =            2      π              3            ∫    0    2                      3                    x        2                            x        3            +      8          d  x  =            2      π        3              [      2                        x          3                +        8            ]        0    2    =      8    3        (    2    −          2        )    πStep 2I have multiplied and divided by 3 in order to have   3      x    2    d  x so the integral becomes  ∫            d      (              x        3            )                      x        3            +      8        =  ∫  (      x    3    +  8      )          −      1              /            2          d  (      x    3    )  =      1          1      −              1        2                (      x    3    +  8      )          1              /            2        +  C  =  2            x      3        +    8    +  C

The region bounded by y=x/sqrt(x^3+8), the x-axis, and the line x=2 is revolved about the y-axis. Find the volume of the solid generated this way.

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