Question

Find the volume of the solid obtained by rotating the region bounded by the curves y = 4 - 4x² and y = 0 about the x-axis.

Answer 1

The volume of the solid obtained by rotating the region bounded by the curves y = 4 - 4x² and y = 0 about the x-axis is 32/3 units³.

Explanation:

To calculate the volume of the solid, the disk method for rotation around the x-axis is applied. The curves intersect at x = ±1. The lower integration limit is -1, and the upper limit is 1 as the region of interest lies between these x-values. The formula for volume using the disk method is V = π ∫[a, b] (f(x))² dx, where f(x) is the radius of the disk.

By substituting y = 4 - 4x² into the formula, the squared radius becomes r² = (4 - 4x²)². After squaring and integrating from -1 to 1, the volume formula becomes V = π ∫[-1, 1] (4 - 4x²)² dx. Solving this integral gives V = π [32x³/3 - 16x⁵/5 + 4x⁷/7] from -1 to 1.

Upon evaluating this definite integral from -1 to 1, the calculation simplifies to V = 32π/3 - 16π/5 + 4π/7 - (-32π/3 + 16π/5 - 4π/7). After computation, the result is V = 32π/3 + 32π/3 = 64π/3. Simplifying yields V = 32/3 units³.

Answer 2

To find the volume of the solid obtained by rotating the region bounded by the curves y = 4 - 4x² and y = 0 about the x-axis, we can use the method of cylindrical shells. The volume is (32π/3) cubic units.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = 4 - 4x² and y = 0 about the x-axis, we can use the method of cylindrical shells.

1. First, we need to determine the limits of integration. Setting the two equations equal to each other, we find that the intersection points are x = -1 and x = 1.
2. We can now set up the integral to find the volume: V = ∫ (2πx)(4-4x²) dx, with the limits of integration from -1 to 1.
3. Integrating this expression will give the volume of the solid. Evaluating the integral, we find that the volume is (32π/3) cubic units.