# Calculus: Find the Volume of the Region Bounded Above by Parabola Z and Below by Square R

**Find the volume of the region bounded above by parabola Z and below by square R **

z = 16 – x^2 – y^2

R: 0 <= x <= 2, 0 <= y <= 2

Because this is a double integral over a general region problem, the equation is going to look like this:

∫R∫ 16 – x^2 – y^2 dA

Essentially, we’re finding the integral of the region covered by parabola Z within the boundaries given by R.

Here we use the x, y bounds given in R to replace R in the above function. For this problem, it doesn’t matter which order you choose to integrate, so long as the integration bounds match the integration variable (i.e. use the x bounds when integrating x).

∫0-2∫0-2 16 – x^2 – y^2 dxdy => ∫0-2 16x – x^3 – xy^2|0-2 dy

Integrating x gives you:

∫0-2 ^{88}⁄_{3} – 2y^2 dy => 88y/3 + (-2y^3)/3 |0-2

Complete the double integral, you should get the answer:

**Answer: ^{160}⁄_{3}**