Explanation:
Let [tex]f(x) = \sqrt{25x}[/tex] and [tex]g(x) = \frac{x^2}{25}[/tex]. The differential volume dV of the cylindrical shells is given by
[tex]dV = 2\pi x[f(x) - g(x)]dx[/tex]
Integrating this expression, we get
[tex]\displaystyle V = 2\pi\int{x[f(x) - g(x)]}dx[/tex]
To determine the limits of integration, we equate the two functions to find their solutions and thus the limits:
[tex]\sqrt{25x} = \dfrac{x^2}{25}[/tex]
We can clearly see that x = 0 is one of the solutions. For the other solution/limit, let's solve for x by first taking the square of the equation above:
[tex]25x = \dfrac{x^4}{(25)^2} \Rightarrow \dfrac{x^3}{(25)^3} = 1[/tex]
or
[tex]x^3 =(25)^3 \Rightarrow x = \pm25[/tex]
Since we are rotating the functions around the y-axis, we are going to use the x = 25 solution as one of the limits. So the expression for the volume of revolution around the y-axis is
[tex]\displaystyle V = 2\pi\int_0^{25}{x\left(\sqrt{25x} - \frac{x^2}{25}\right)}dx[/tex]
[tex]\displaystyle\:\:\:\:=10\pi\int_0^{25}{x^{3/2}}dx - \frac{2\pi}{25}\int_0^{25}{x^3}dx[/tex]
[tex]\:\:\:\:=\left(4\pi x^{5/2} - \dfrac{\pi}{50}x^4\right)_0^{25}[/tex]
[tex]\:\:\:\:=4\pi(3125) - \pi(7812.5) = 14726.2[/tex]
