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: