Question

Find the volume generated by revolving the given region about the given axis.

(a) The region bounded by y = x⁴ , x = 1 and y = 0 about Y axis.

(b) The triangle with vertices (1, 1),(1, 2)(2, 2) about X axis.

(c) The region in the first quadrant bounded by x = y − y³ , x = 1 and y = 1 about X axis. 

Answer 1

(a)

$$A_1(y)=\pi x^2=\pi\sqrt{y}$$

$$A_2(y)=\pi(x)^2=\pi$$

$$V=\displaystyle\int_{0}^{1}(\pi-\pi\sqrt{y})dy=\pi[y-\dfrac{2}{3}y^{3/2}]\begin{matrix} 1 \\ 0 \end{matrix}$$

$$=\dfrac{\pi}{3} (cubic\ units)$$

(b)

$$V=\displaystyle\int_{1}^{2}(\pi(2)^2-\pi x^2)dy=\pi[4x-\dfrac{1}{3}x^{3}]\begin{matrix} 2\\ 1 \end{matrix}$$

$$=\pi(8-\dfrac{8}{3}-(4-\dfrac{1}{3})=\dfrac{5}{3}(cubic\ units)$$

(c)

$$V=\displaystyle\int_{0}^{1}2\pi y(1-(y − y^3))dy=2\pi[\dfrac{y^2}{2}-\dfrac{y^3}{3}+\dfrac{y^4}{4}]\begin{matrix} 1\\ 0 \end{matrix}$$

$$=\dfrac{5\pi}{12}(cubic\ units)$$