Question

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = sqrt(25x) and y = x^2/25. Find V by slicing & find V by cylindrical shells.

Answer 1

Step-by-step explanation:

Let
`f(x) = √(25x)` and
`g(x) = (x^2)/(25)`. The differential volume dV of the cylindrical shells is given by

`dV = 2\pi x[f(x) - g(x)]dx`

Integrating this expression, we get

`\displaystyle V = 2\pi\int{x[f(x) - g(x)]}dx`

To determine the limits of integration, we equate the two functions to find their solutions and thus the limits:

`√(25x) = (x^2)/(25)`

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:

`25x = (x^4)/((25)^2) \Rightarrow (x^3)/((25)^3) = 1`

or

`x^3 =(25)^3 \Rightarrow x = \pm25`

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

`\displaystyle V = 2\pi\int_0^(25){x\left(√(25x) - (x^2)/(25)\right)}dx`

`\displaystyle\:\:\:\:=10\pi\int_0^(25){x^(3/2)}dx - (2\pi)/(25)\int_0^(25){x^3}dx`

`\:\:\:\:=\left(4\pi x^(5/2) - (\pi)/(50)x^4\right)_0^(25)`

`\:\:\:\:=4\pi(3125) - \pi(7812.5) = 14726.2`