Question

Let V be the volume of the solid S obtained by rotating about the y-axis the region bounded by y=√(72 x) and y=3 x². Find V

Answer 1

The volume
`\( V \)`of the solid
`\( S \)` obtained by rotating the region bounded by
`\( y = √(72x) \) and \( y = 3x^2 \) about the y-axis is \( V = (288)/(5) \pi \)` cubic units.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the given region about the y-axis, we can use the disk method. The limits of integration are determined by finding the points where the two curves intersect. Setting
`\( √(72x) = 3x^2 \), we find \( x = (4)/(3) \) and \( x = 6 \)`. These are the limits of integration.

Now, consider an elemental disk at a distance
`\( x \)` from the y-axis. The radius of this disk is
`\( y \)`, which varies from
`\( √(72x) \) to \( 3x^2 \).` The volume
`\( dV \)` of this disk is given by
`\( dV = \pi y^2 dx \).`

Integrating
`\( dV \) from \( x = (4)/(3) \) to \( x = 6 \)` gives the total volume
`\( V \): \[ V = \int_{(4)/(3)}^(6) \pi (√(72x))^2 - (\pi (3x^2)^2) \,dx \]`

Solving this integral yields
`\( V = (288)/(5) \pi \)` cubic units, which is the final answer.

In summary, by setting up the integral using the disk method and evaluating it within the given limits, we find the volume of the solid obtained by rotating the specified region about the y-axis is
`\( (288)/(5) \pi \)`cubic units.