Question

find the volume of the solid that lies under the hyperbolic paraboloid z=3y²−x²+2 and above the rectangle r=[−1,1]×[1,2]

Answer 1

To find the volume of the solid that lies under the hyperbolic paraboloid z=3y²−x²+2 and above the rectangle r=[−1,1]×[1,2], we can use a double integral.

Step-by-step explanation:

To find the volume of the solid that lies under the hyperbolic paraboloid z=3y²−x²+2 and above the rectangle r=[−1,1]×[1,2], we can use a double integral. The volume is given by the integral of the function z=3y²−x²+2 over the rectangle r. We integrate this function with respect to x and y over the given bounds of the rectangle and then multiply by the area of the rectangle.

The volume V is given by the double integral:

V = ∫∫r (3y²−x²+2) dA

where dA represents the area element in the xy-plane. By evaluating the integral, we can find the volume of the solid.