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 y = 5, x = 9 − (y − 2)2

Answer 1

The volume V of the solid obtained by rotating the region about the x-axis is V = 420π cubic units.

Step-by-step explanation:

To find the volume using cylindrical shells, we'll integrate along the y-axis because the curves are given as functions of y. First, we need to determine the limits of integration by finding the points of intersection between the curves x = 5/y and x = 9 - (y - 2)^2.

Setting the equations equal gives us: 5/y = 9 - (y - 2)^2. After simplification and solving, we find the limits of integration as y = 1 and y = 3.

Next, we establish the radius and height of the shells. The radius is the distance from the axis of rotation (x-axis) to the curve x = 5/y, which is simply x = 5/y. The height of each shell is the difference between the equations: x = 9 - (y - 2)^2 and x = 5/y, so it becomes h = 9 - (y - 2)^2 - 5/y.

The volume V is then calculated by integrating the product of the circumference (2πr) and height (h) with respect to y from y = 1 to y = 3:

V=
`\int\limits^1_3`2π⋅( 5)⋅(9−(y−2) 2− y5)dy.

Solving this integral will give the volume of the solid of revolution. Upon evaluation, V = 420π cubic units.

This method of cylindrical shells involves slicing the solid perpendicular to the axis of rotation, forming cylindrical shells that approximate the volume of the solid. Integrating these shells over the specified interval of y-values gives the total volume of the solid generated by rotating the region about the x-axis.