Question

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 25x² - 25, y = 0, x = 0, and x = 5.

Answer 1

To find the volume of the solid obtained by rotating about the x-axis the region enclosed by the given curves, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the given region about the x-axis, we can use the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = 2πx(f(x)-g(x))dx, where f(x) and g(x) are the equations of the upper and lower curves, and x is the variable of integration.

For this problem, the upper curve is y = 25x² - 25 and the lower curve is y = 0. To set up the integral, we need to express y in terms of x for both curves.

The integral setup would be: V = ∫(from 0 to 5) of 2πx((25x² - 25) - 0)dx. Solving the integral will give us the volume of the solid.