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After Rotating to about y-axis

The total volume of solid is

\(V = \int dv = 2\pi \int_a^b x f(x) dx\)

Then in this case, we have

a = 0

b = 1

The volume of given solid is

\(V = \int dv = 2\pi \int_a^b x (y_1 – y_2) dx\)

\(V = 2\pi \int_{0}^{1}x(x – x^2) dx\)

\(= 2\pi \int_{0}^{1}(x^2 – x^3)\)

\(= 2\pi \left [ \frac{x^3}{3} – \frac{x^4}{4} \right ]_{0}^{1}\)

\(= 2\pi \left ( \frac{1}{3} – \frac{1}{4} \right )\)

\(= \frac{\pi}{6} units\)

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