How do you find the volume of the solid generated by revolving the region bounded by the curves y = 10 / x², y = 0, x = 1, x = 5 rotated about the y-axis?
Integrate by Method of Rings
(1) Determine the plot of
(2) In this solution, the positive side of the curve was used (so as not to deal with negative signs ^_^).
(3) Since the curve is to be rotated about the y-axis, the cross section of the solid should be perpendicular to the y-axis and has its area a function of y.
(4) Since the curve is to be bounded from x=1, the inner radius of the ring or the distance of the line x=1 to the axis of rotation (which is x=0) is equal to 1 .
(5) As for the outer radius of the ring, the distance of the curve to the axis of rotation is expressed as
(6) Hence the area of the ring is,
(7) If we take a differential element, dy, multiply it to the cross sectional area, then integrate it, we get the volume of the solid. As for the limits of integration, find the values of y at x = 1 and x = 5 based on the curve
(8) Determining the volume,