Question

Set up integrals for the volumes of the given solids. (But don't evaluate them.)

(a) The region bounded by x = −2y² , x = y² −3y, z = y² , and z = 0.

Answer 1

To set up the integrals for the volumes of the given solids, we first need to find the limits of integration for each variable. We have x = -2y² and x = y² - 3y, so the limits for x are the values where these two equations intersect. The limits for y are the values that satisfy z = y² and z = 0. Finally, the limits for z are already given.

Step-by-step explanation:

To set up integrals for the volumes of the given solids, we need to find the limits of integration for each variable. Let's start with x. We have x = -2y² and x = y² - 3y, so the limits for x are the values where these two equations intersect. Solving for y, we get y = 0 and y = 2. Now let's look at y. The limits for y are the values that satisfy z = y² and z = 0. Solving for y, we get y = 0 and y = 1. Finally, the limits for z are already given as z = y² and z = 0. Therefore, the integrals for the volumes of the solids are:

• For the solid between x = -2y² and x = y² - 3y, the integral is: ∫02 ∫01 ∫y²0 dz dy dx
• For the solid between x = y² - 3y and x = -2y², the integral is: ∫02 ∫01 ∫0y² dz dy dx