Question

Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. The equations are y = 5x² and y = 0. The line of revolution is x = 2. What is the volume of the solid generated?

Answer 1

To find the volume of the solid generated by revolving the region bounded by the graphs of the equations y = 5x² and y = 0 about the line x = 2, we can use the method of cylindrical shells. The volume of the solid generated is 80π cubic units.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the graphs of the equations y = 5x² and y = 0 about the line x = 2, we can use the method of cylindrical shells.

First, we need to find the limits of integration. Since the line x = 2 is the axis of rotation, the limits will be from x = 0 to x = 2.

Next, we need to express the equation y = 5x² in terms of x. Solving for x, we get x = ±√(y/5).

Now, we can find the volume using the formula V = 2π∫(x)(y)dx, where x goes from 0 to 2 and y goes from 0 to 5x².

Integrating, we get V = 2π∫(x)(5x²)dx = 2π∫5x³dx = 2π(5/4)x⁴|₀² = 20π(16/4 - 0) = 80π.

The volume of the solid generated is 80π cubic units.