Use cylindrical shells to find the volume of the solid generated when the region R under y = x³ over the interval [0, 3] is revolved about the x-axis.

Question

Quizplus · Accepted Answer

We can use cylindrical shells to find the volume of the solid generated by revolving the region R about the x-axis. The formula for the volume of a cylindrical shell is:

V = 2πrh * Δx

where r is the radius of the shell, h is the height of the shell, and Δx is the thickness of the shell.

To use cylindrical shells, we need to express y = xS1U1P13S1S1P0 in terms of x. Since the function involves three different functions, we will split the interval [0, 3] into three subintervals, each corresponding to a different function.

For the first subinterval, 0 ≤ x ≤ 1, we have:

y = xS1 = x

For the second subinterval, 1 ≤ x ≤ 3^(1/3), we have:

y = xU1P13S1S1P0 = x^(1/3)

For the third subinterval, 3^(1/3) ≤ x ≤ 3, we have:

y = xP0 = 1

Now we can set up the integral to find the volume:

V = ∫(from 0 to 1) 2πx*x * dx + ∫(from 1 to 3^(1/3)) 2πx*x^(1/3) * dx + ∫(from 3^(1/3) to 3) 2πx*1 * dx

Simplifying each integral and evaluating them, we get:

V = π/2 + 2π/5(27^(5/3) - 1) + 9π/2

This simplifies to:

V = (73π)/5

Therefore, the correct answer is B.

Use Cylindrical Shells to Find the Volume of the Solid