# Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.

Solution:

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

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

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

$$= 2\pi \int_{0}^{1}(x^2 – x^3)$$

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

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

$$= \frac{\pi}{6} units$$