Finding volume of solid revolution about x-axisI'm asked to find the volume of a container of height 13.5 The container is made by rotating y = 1.5 x 2 where 0 ≤ x ≤ 3 about the axis x = − 0.5 (the bottom is flat).I tried finding the right volume a number of ways, but I believe my most solid attempt so far is this:I invert the function to get y = 2 3 x .From this I tried to calculate π ∗ ∫ 0 3 2 3 x d x which gave me 3 π which I thought was correct, but it wasnt. Plotting this, I realized the height becomes 3 when I integrate the inverted function from 0 to 3, so I thought I should maybe integrate from 0 to 13.5. In that case I get 243 π 4 which is also incorrect. I'm sure I've misinterpreted what it means to rotate the function about the axis x = − 0.5

Question

Finding volume of solid revolution about x-axisI&#039;m asked to find the volume of a container of height 13.5 The container is made by rotating   y  =  1.5      x    2   where   0  ≤  x  ≤  3 about the axis   x  =  −  0.5 (the bottom is flat).I tried finding the right volume a number of ways, but I believe my most solid attempt so far is this:I invert the function to get   y  =            2      3        x  .From this I tried to calculate   π  ∗      ∫          0              3                  2      3        x    d  x which gave me   3  π which I thought was correct, but it wasnt. Plotting this, I realized the height becomes 3 when I integrate the inverted function from 0 to 3, so I thought I should maybe integrate from 0 to 13.5. In that case I get             243      π        4   which is also incorrect. I&#039;m sure I&#039;ve misinterpreted what it means to rotate the function about the axis   x  =  −  0.5

kunstboom8w · Accepted Answer

Explanation:I translated 0.5 units to the right so I had the equation   y  =      3    2              (      x      −              1        2            )        2   to rotate around y axis and I got   V  =  π      ∫    0    3              (              3        2                              (          x          −                      1            2                    )                2            )        2      d  x  =            14067      π        320    ≈  138.1

The container is made by rotating y=1.5x^{2} where 0 leq x leq 3 about the axis x=-0.5 (the bottom is flat).

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