Question

Volume of the Solid (Shell Method)

Given the bounded region is revolved about y axis, find the volume of the solid generated.

The region below the curve y= Inx, above the x-axis, and to the left of the line x=4.

Answer 1

$$V=2\pi\displaystyle\int_{0}^4x\ln xdx$$

$$\int x\ln x dx$$

$$u=\ln x, du=\dfrac{dx}{x}$$

$$dv=xdx, v=\dfrac{x^2}{2}$$

$$\int x\ln x dx=\dfrac{x^2\ln x}{2}-\dfrac{1}{2}\int xdx$$

$$=\dfrac{x^2\ln x}{2}-\dfrac{x^2}{4}+C$$

$$V=2\pi\displaystyle\int_{0}^4x\ln xdx$$

$$=2\pi\lim\limits_{t\to 0^+}\displaystyle\int_{t}^4x\ln xdx$$

$$=2\pi\lim\limits_{t\to 0^+}[\dfrac{x^2\ln x}{2}-\dfrac{x^2}{4}]\begin{matrix} 4 \\ t \end{matrix}$$

$$=2\pi(\dfrac{4^2\ln 4}{42}-\dfrac{4^2}{4}-(0-0))$$

$$=8\pi(4\ln 2-1)\ ({units^3})$$