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Math Help - volumes rotate about y axis

  1. #1
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    volumes rotate about y axis

    The region of the curve  y = \frac{e^x}{x} bounded by x = 1 and x = 2 and y = 0, is rotated about the y axis. Find the volume of the solid.

    i drew the diagram and it doesn't look doable. Would I have to consider volumes by slicing?

    thanks
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  2. #2
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    It certainly is "doable". The region in the xy-plane is a "quadrilateral with one curved side, vertices at (1, 0), (2, 0), (2, e^4/2) and (1, e). There would be a problem with doing it as "washers" since there would be a change of formula at (1, e).

    If you "slice" it parallel to the y-axis, rotating around the y-axis, each "slice" would be a cylinder, of thickness dx, and height the distance from y= 0 to y= e^x/x which is e^x/x.

    Since this is rotated around the x-axis, the radius of each cylinder is x and the area is \pi r^2h= \pi x^2(e^x/x)= \pi xe^x. The volume of each thin cylinder is \pi x e^x dx and the entire volume is given by the integral
    \pi \int_1^2 x e^x dx.
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