Question

Use shells to find the volume generated by rotations the regions between the given curve and the y-axis around the x-axis: (a) x= 1+y 2 1 ​ ,y=1 and y=4; (b) y=x 2 ,y=0 and y=2.

Answer 1

(a) The volume generated by rotating the region between the curve x = 1 + y^2, y = 1, and y = 4 around the x-axis is
`\((154π)/(15)\)`. (b) For the region between the curves y = x^2, y = 0, and y = 2 rotated around the x-axis, the volume is
`\((32π)/(5)\).`

Step-by-step explanation:

In part (a), we employ the shell method to determine the volume. The integral for the volume V is given by:

`\[V = \int_(1)^(4) 2π(1 + y^2)(4 - y) \, dy\]`

The factor of 2π accounts for the cylindrical shape of the shells, while
`(1 + y^2)`represents the radius of the shell, and (4 - y) is the height. Evaluating this integral yields the final answer
`\((154π)/(15)\)`.

For part (b), the integral for the volume V is set up as follows:

`\[V = \int_(0)^(2) 2πx(x^2) \, dx\]`

Here, 2π represents the cylindrical surface area, x is the radius, and
`x^2` is the height. Solving this integral gives the result
`\((32π)/(5)\).` This demonstrates the volume of the solid formed by rotating the given region about the x-axis.

These calculations illustrate the application of the shell method, where the integral is structured based on the cylindrical shells perpendicular to the axis of rotation.

The resulting volumes provide a comprehensive understanding of the spatial configuration generated by the rotation of the specified regions.