Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y = 3x2, y = 18x − 6x2

Answer 1

To find the volume generated by rotating the region bounded by the given curves about the y-axis using the method of cylindrical shells, follow these steps: Determine the interval of integration, set up the integral using the volume formula for a cylindrical shell, and integrate the integral over the interval of integration.

Step-by-step explanation:

To find the volume generated by rotating the region bounded by the given curves about the y-axis using the method of cylindrical shells, we need to consider the height, radius, and thickness of each cylindrical shell.

Step 1: Determine the interval of integration by finding the intersection points of the two curves, which are the values of x where y = 3x^2 and y = 18x - 6x^2 are equal.

Step 2: Set up the integral by considering a vertical slice of the region and using the formula for the volume of a cylindrical shell, which is V = 2πrh*Δx, where r is the radius, h is the height, and Δx is the thickness of the shell.

Step 3: Integrate the integral from step 2 over the interval of integration to find the total volume.

Answer 2

The method of cylindrical shells computes the volume by integrating over cylindrical 'shells' created by rotating the region between two functions around an axis. The volume of each shell is given by V = 2πrh, and integrating this across the bounds set by the intersections of the curves gives the total volume.

Step-by-step explanation:

The method of cylindrical shells involves integrating the volume of thin cylindrical 'shells' to find the total volume of a solid of revolution. To calculate the volume generated by rotating the region bounded by the curves y = 3x2 and y = 18x - 6x2 around the y-axis, we need to follow these steps:

Sketch the region to identify the bounds of integration.

Set up the integral for the volume using the formula for cylindrical shells, which combines the perimeter of the base circle (circumference), the height of the cylinder (the difference in the functions), and the shell's thickness.

Perform the integration with the appropriate limits to obtain the volume.

Each cylindrical shell's volume is given by V = 2πrh, where r is the radius (distance from the y-axis to the shell) and h is the height (the value of the upper function minus the lower function at a particular x).