Question

use the method of cylindrical shells to find the volume v of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 10y2 − 2y3, x = 0

Answer 1

`\[ V = (200\pi)/(3) \]`

Step-by-step explanation:

To find the volume using the method of cylindrical shells, we integrate the circumference of the shells multiplied by their height. The region is rotated about the x-axis between
`\(x = 0\)`and the curve
`\(x = 10y^2 - 2y^3\).`The height of the shell is
`\(dy\)`(infinitesimally small change in
`\(y\)),` and the radius is
`\(x\)`. The circumference is
`\(2\pi x\),` and the height is
`\(dy\)`. The integral is set up as follows:

`\[ V = \int_(0)^(1) 2\pi x \, dy \]`
`\(x = 0\)`

Substitute
`\(x = 10y^2 - 2y^3\)` into the integral:

`\[ V = \int`
`_(0)^(1) 2\pi (10y^2 - 2y^3) \, dy \]`

Simplify the expression and perform the integration:

`\[ V = (200\pi)/(3) \]`

The final answer,
`\((200\pi)/(3)\)`, represents the volume of the solid obtained by rotating the region about the x-axis. This result is obtained by evaluating the definite integral that considers the cylindrical shells formed by the rotation of the given region. The integration process involves expressing the circumference of the shells, given by
`\(2\pi x\)`, in terms of the variable
`\(y\)`and integrating with respect to
`\(y\)` over the specified interval. The integral simplifies to the provided expression, yielding the volume of the solid.