Question

The volume of the solid obtained by rotating the region bounded by y = x², and y = 5x about the line x = 5 can be computed using either the washer method or the method of cylindrical shells. Answer the following questions. Using the washer method, set up the integral. V = with a = and b = Using the method of cylindrical shells, set up the integral. V = with c = and d = Choose either integral to find the volume.

Answer 1

To find the volume using the washer method, set up the integral V = ∫(a to b) π[(5x)² - (x²)²] dx. To find the volume using the method of cylindrical shells, set up the integral V = ∫(c to d) 2π(5 - x)[(5x) - x²] dx.

Step-by-step explanation:

To set up the integral using the washer method, we need to consider the cross-sections of the solid obtained by rotating the region bounded by y = x² and y = 5x about the line x = 5. The cross-sections are washers with outer radius equal to 5x and inner radius equal to x². The height of each washer is determined by the difference between the functions y = x² and y = 5x. Therefore, the integral to find the volume is:

V = ∫a b π[(5x)² - (x²)²] dx

To set up the integral using the method of cylindrical shells, we need to consider the vertical shells formed by rotating the region bounded by y = x² and y = 5x about the line x = 5. Each shell has a radius of 5 - x and a height determined by the difference between the functions y = x² and y = 5x. Therefore, the integral to find the volume is:

V = ∫c d 2π(5 - x)[(5x) - x²] dx