Question

8. Find the volume of solid formed by revolving a loop of the lemniscate r^2 = a^2 cos 20 about
initial line.​

Answer 1

My first thought was to take slices perpendicular to the line y=xy=x of radius

∣∣x−y2√∣∣|x−y2| (perpendicular distance of point to line y=x).

However the cartesian equation of the lemniscate is (x2+y2)2=a2(y2−x2)(x2+y2)2=a2(y2−x2), which I can't seem to express nicely in terms of yy so that I can calculate the volume using an integral with respect to xx.

I then thought of rotating the lemniscate π4π4 clockwise to the curve r2=a2cos2θr2=a2cos⁡2θ and using cylindrical shells or slices but again I ran into the problem of trying to express yy in terms of xx (using the quadratic formula gave me the square root of a quartic which doesn't seem integrable).

I know there are approximations I can use, but I just want to know if this problem is possible with only these basic methods (washers/slices and cylindrical shells) and not Pappus's centroid theorem which I have learnt yet.

Hope it helps..

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