Question

Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by y=2x^2 and y=x^3 about the x-axis.

Answer 1

$$2x^2=x^3$$

$$x_1=0, x_2=2$$

$$V=\pi\displaystyle\int_{0}^{2}\big((2x^2)^2-(x^3)^2\big)dx$$

$$=\pi\big[\dfrac{4x^5}{5}-\dfrac{x^7}{7}\big]\begin{matrix} 2 \\ 0 \end{matrix}=\pi(\dfrac{4(2)^5}{5}-\dfrac{(2)^7}{7}-0)$$

$$=\dfrac{256\pi}{35}({units}^3)$$