Question

Find the volume of the region between the cylinder z=4y²?

Answer 1

To find the volume of the region between the cylinder z=4y², integrate the function over the given region. Set up the integral by finding the limits of integration for y and then integrating the function with respect to y. The volume of the region is 4/3 cubic units.

Step-by-step explanation:

To find the volume of the region between the cylinder z=4y², we need to integrate the function over the given region. In this case, the region is defined by the equation z=4y². Since this equation is only in terms of y, we can integrate with respect to y. Let's set up the integral:

1. First, find the limits of integration for y by setting z=0 and solving for y:

   0 = 4y²

   y = 0

2. Next, integrate the function:

   V = ∫(0 to 1) 4y² dy

   V = 4∫(0 to 1) y^2 dy

   V = 4[y^3/3] (0 to 1)

   V = 4(1/3 - 0)

   V = 4/3

Therefore, the volume of the region between the cylinder z=4y² is 4/3 cubic units.