Question

Find the volume of revolution formed by rotating the curve y = x + x², the x -axis and the ordinates x = 2and x = 3.

Answer 1

Given

Find the volume of revolution formed by rotating the curve y = x + x², the x -axis and the ordinates x = 2

and x = 3.

Solution

The formula

`The\text{ volume of revolution =}\int ^b_a\pi y^2dx`

Given

`\begin{gathered} y=x+x^2 \\ y^2=(x+x^2)^2 \\ y^2=x^2+2x^3+x^4 \end{gathered}`
`\begin{gathered} The\text{ volume of revolution =}\int ^b_a\pi y^2dx \\ The\text{ volume of revolution =}\int ^b_a\pi(x^2+2x^3+x^4)^{}dx \\ a=2 \\ b=3 \\ \\ The\text{ volume of revolution =}\int ^3_2\pi(x^2+2x^3+x^4)^{}dx \end{gathered}`
`\begin{gathered} \int ^3_2\pi(x^2+2x^3+x^4)^{}dx=\pi\int ^3_2x^2+2x^3+x^4dx \\ \\ Apply\text{ sum rule} \\ \pi\mleft(\int ^3_2x^2dx+\int ^3_22x^3dx+\int ^3_2x^4dx\mright) \\ \\ \int ^3_2x^2dx=(19)/(3) \\ \\ \int ^3_22x^3dx=(65)/(2) \\ \\ \int ^3_2x^4dx=(211)/(5) \end{gathered}`
`\begin{gathered} \pi\mleft((19)/(3)+(65)/(2)+(211)/(5)\mright) \\ \text{Simplify the bracket} \\ =\pi(2431)/(30) \end{gathered}`

The final answer

`\text{The volume of the revolution }=(2431)/(30)\pi`