Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8?

Answer 1

The volume V generated by rotating the region bounded by the curves
`\( y = √(x) \) and \( y = x^2 \) about \( y = 8 \)` using the method of cylindrical shells i
`s \( V = (15488)/(15) \)` cubic units.

Step-by-step explanation:

To find the volume using cylindrical shells, we integrate the product of the circumference of a shell the height of the shell and the thickness of the shell. The region between
`\( y = √(x) \) and \( y = x^2 \)`is revolved about
`\( y = 8 \)`. To set up the integral, consider a shell at height y with radius
`\( R(y) \)`and thickness
`\( \Delta y \).` The circumference of the shell is ( 2pi R(y) the height is
`( \Delta y \),`and the thickness is
`\( \Delta y \)`.

Now R(y) is the distance from the axis of rotation y = 8 to the outer curve
`\( y = x^2 \), so \( R(y) = 8 - x^2 \)`. Substitute
`\( x = √(y) \) to get \( R(y) = 8 - y \)`. The volume element is
`\( dV = 2\pi(8-y)\Delta y \cdot \Delta y \)`. The integral becomes
`\( V = \int_(0)^(4) 2\pi(8-y)y\,dy \).` Simplifying this integral yields the final answer
`\( V = (15488)/(15) \).`

In conclusion, the volume V is calculated by integrating the product of the circumference, height, and thickness of each cylindrical shell. The integral is set up using the outer curve
`\( y = x^2 \)`as the radius function and the final result is
`\( (15488)/(15) \)` cubic units.