Cubic Rotation

Calculus Level 3

The area bounded by the curve y = 2 x 2 x 3 y = 2x^2-x^3 and line y = 0 y=0 is rotated around the y-axis. What is the volume of the resulting structure?


The answer is 10.053096.

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1 solution

A K
May 2, 2014

Such problems can be solved with the general formula:

V o l u m e = 2 π a b x f ( x ) d x Volume = 2\pi \int_a^b xf(x)\,\mathrm{d}x

Factorising shows that the region in question is between x=0 and x=2, hence a = 0 a=0 and b = 2 b=2 .

Therefore:

V o l u m e = 2 π 0 2 x ( 2 x 2 x 3 ) d x = 16 π 5 Volume = 2\pi \int_0^2 x(2x^{2}-x^{3})\,\mathrm{d}x = \boxed{\frac{16\pi}{5}}

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