Find the volume formed by revolving about line 𝑥 = 1 the area bounded by the curve 𝑦 = (𝑥2 −1)2 and the x-axis.
To find the volume formed by revolving the given area about the line 𝑥 = 1, we can use the method of cylindrical shells.
First, let's analyze the problem. The given curve 𝑦 = (𝑥^2 − 1)^2 is symmetric about the y-axis. We want to find the volume when this curve is revolved about the line 𝑥 = 1.
To do this, we'll consider an infinitesimally thin vertical strip (or shell) oriented along the x-axis. The height of each shell will be the y-coordinate of the curve, and the radius will be the distance from 𝑥 = 1 to the x-coordinate of the curve.
Let's break down these steps into smaller parts to find the volume:
Step 1: Determine the bounds of integration.
We need to find the x-values at which the given curve intersects the x-axis. To do this, we set 𝑦 = 0 and solve for 𝑥:
(𝑥^2 − 1)^2 = 0
Taking the square root of both sides, we get:
𝑥^2 − 1 = 0
Simplifying, we find:
𝑥^2 = 1
Taking the square root again, we get the two x-values of intersection:
𝑥 = ±1
Therefore, the bounds of integration are 𝑥 = -1 to 𝑥 = 1.
Step 2: Determine an expression for the radius.
Since we are revolving the curve about the line 𝑥 = 1, the radius of each shell is given by:
radius = 𝑥 - 1
Step 3: Determine an expression for the height.
The height of each shell is given by the y-coordinate of the curve. In this case, the y-coordinate is given by:
𝑦 = (𝑥^2 − 1)^2
So, the height for each shell is:
height = (𝑥^2 − 1)^2
Step 4: Set up the integral.
Using the formula for the volume of a cylindrical shell, the volume of the solid revolution is:
V = ∫[a, b] 2π(radius)(height) 𝑑𝑥
Since our bounds are 𝑥 = -1 to 𝑥 = 1, the integral becomes:
V = ∫[-1, 1] 2π(𝑥 - 1)(𝑥^2 − 1)^2 𝑑𝑥
Now, we can evaluate this integral to find the volume.
Note: Calculating the integral may involve some complicated algebraic manipulations or the use of software.