Question

Find the volume of the solid obtained by rotating the region bounded by the curves y = 8 - 8x² and y = 0 about the x-axis.

Answer 1

The volume of the solid formed by rotating the region between the curves y = 8 - 8x² and y = 0 about the x-axis is found using the disk method, setting up an integral from -1 to 1, and then evaluating it.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the given curves around the x-axis, we can use the disk method. The curves provided are y = 8 - 8x² and y = 0, which intersect when y = 8 - 8x² = 0, giving the x-values for the bounds of integration.

First, we solve for x to find the bounds:

• 8 - 8x² = 0
• x² = 1
• x = ±1

Next, we set up the integral for volume using the disk method:

• V = π ∫_{-1}^{1} (8 - 8x²)² dx

We then evaluate the integral, which will give us the volume of the solid of revolution. Evaluating this integral requires algebraic manipulation and integration techniques, which would be done through an integral solver for this explanation.