Question

Find the volume of the solid that lies under the hyperbolic paraboloid z = 20 + x² − y² and above the rectangle. R = [−2, 2] × [0, 4]

Answer 1

The volume of the solid under the hyperbolic paraboloid z=20+x²-y² and above the rectangle R=[-2,2] × [0,4] is calculated using a double integral over the specified ranges for x and y, and integrating the given function.

Step-by-step explanation:

To find the volume of the solid that lies under the hyperbolic paraboloid z = 20 + x² − y² and above the rectangle R = [−2, 2] × [0, 4], we can use a double integral. The volume V is given by:

V = ∫∫_R (20 + x² - y²) dA

Here, dA is the differential area element of the rectangle R in the xy-plane.

V = ∫_{y=0}^{y=4} ∫_{x=-2}^{x=2} (20 + x² - y²) dx dy

This integral can be computed as follows:

1. Integrate with respect to x:

2. We compute the integral of (20 + x²) from x = −2 to x = 2, and subtract the integral of y² over the same interval.

3. Integrate the resulting expression with respect to y from y = 0 to y = 4.

After performing these integrations, we will obtain a numerical value that represents the volume of the solid.