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
[tex]The\text{ volume of revolution =}\int ^b_a\pi y^2dx[/tex]
Given
[tex]\begin{gathered} y=x+x^2 \\ y^2=(x+x^2)^2 \\ y^2=x^2+2x^3+x^4 \end{gathered}[/tex][tex]\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}[/tex][tex]\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=\frac{19}{3} \\ \\ \int ^3_22x^3dx=\frac{65}{2} \\ \\ \int ^3_2x^4dx=\frac{211}{5} \end{gathered}[/tex][tex]\begin{gathered} \pi\mleft(\frac{19}{3}+\frac{65}{2}+\frac{211}{5}\mright) \\ \text{Simplify the bracket} \\ =\pi\frac{2431}{30} \end{gathered}[/tex]
The final answer
[tex]\text{The volume of the revolution }=\frac{2431}{30}\pi[/tex]
