Find the volume of the solid formed by revolving the region bounded by the graphs of y = x³ + x² and y = 1 about the y-axis

To find the volume of the solid formed by revolving the region between the curves y = x³ + x² and y = 1 about the y-axis, we can use the method of cylindrical shells.

First, we need to determine the points of intersection between the two graphs. We set the equations equal to each other:

x³ + x² = 1

Now we can rearrange it:

x³ + x² - 1 = 0

This cubic equation can be solved numerically or graphically to find the x-values where the curves intersect. Using numerical methods or graphing tools, we find that the real root of the equation lies between 0 and 1. Let’s denote this intersection point as x = a.

Next, we use the formula for the volume using cylindrical shells:

V = 2π ∫[a to b] (radius)(height) dx

In our case, the radius is x and the height of the shell is 1 – (x³ + x²). Therefore, we can express the volume as:

V = 2π ∫[0 to a] x(1 - (x³ + x²)) dx

Now, simplifying the integrand gives:

V = 2π ∫[0 to a] (x - x⁴ - x³) dx

Calculating this integral requires integrating the terms:

= 2π [(1/2)x² - (1/5)x⁵ - (1/4)x⁴] |[0 to a]

After evaluating the integral from 0 to a, we plug in the value of a, which we had previously found as the intersection point. This will yield the volume of the solid formed by the revolution.

To summarize, the key steps to find the volume involved determining the points of intersection, setting up the integral based on the dimensions of the shell, and finally evaluating that integral using the proper limits. This gives us the volume of the solid.