Question

Find the volume of the solid that lies under the hyperbolic paraboloid z = 19 + x2 − y2 and above the rectangle. R = [−2, 2] × [0, 4].

Answer 1

Answer:

44 cubic units

Step-by-step explanation:

we are given that a solid lies under the hyperbolic paraboloid

$z = 19 + x^2 - y^2$

and above the rectangle. R = [−2, 2] × [0, 4].

Thus we see in xy plane x varies from -2 to 0, y varies from 2 to 4 while z varies from 0 to $19 + x^2 - y^2$

To find volume we use triple integrals.

$V=\int_2^4 \int_{-2} ^0 \int_0^{19 + x^2 - y^2} dzdxdy\\= \int_2^4 \int_{-2} ^0 (19 + x^2 - y^2} dxdy\\= \int_2^4 19x+\frac{x^3}{3} -xy^2 dy \\=\int_2^4 38-2y^2+\frac{8}{3} dy\\=38y-\frac{2y^3}{3} +\frac{8y}{3}\\=44$

Using triple integral we find volume of the solid is equal to 44 cubic units.