Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = 32 − x^2, y = x^2; about x = 4
To find the volume generated by rotating the region bounded by the curves y = 32 − x² and y = x² about the line x = 4 using the method of cylindrical shells, follow these steps:
Step 1:
First, sketch the region bounded by the given curves. This will help visualize the shape we'll be rotating.
Step 2:
Determine the limits of integration. In this case, we need to find the x-values at which the curves intersect. By solving the equations y = 32 − x² and y = x² simultaneously, we can find these points.
Setting the two equations equal to each other, we have:
32 - x² = x²
Solving for x, we get:
2x² = 32
x² = 16
x = ±√16
x = ±4
So, the region is bounded by x = -4 and x = 4.
Step 3:
To set up the integral for the volume, we need to find the height and radius of each cylindrical shell.
The height is given by the difference between the y-values of the two curves at a particular x-value, and can be calculated as follows:
h = (32 - x²) - x²
h = 32 - 2x²
The radius is the distance between the axis of rotation (x = 4) and the x-value of each cylindrical shell, given by:
r = 4 - x
Step 4:
Now, we can set up the integral for the volume V. The volume of each cylindrical shell is given by:
dV = 2πrh dx
To find the total volume, we integrate the expression 2πrh with respect to x over the limits of integration:
V = ∫(from -4 to 4) 2π(4 - x)(32 - 2x²) dx
Step 5:
Evaluate the integral using the limits of integration and solve for V.
By simplifying the integrand and performing the integration, we get:
V = 2π ∫(from -4 to 4) (128 - 80x² + 2x⁴) dx
After integrating and evaluating the limits of integration, you will find the volume V generated by rotating the region about the line x = 4.
yeah, "James" ...
v = ∫[0,4] 2πrh dx
where r = 4-x and h = (32-x^2)-(x^2)
v = ∫[0,4] 2π(4-x)(32-2x^2) dx = 1280π/3