# Find the volume of the solid under the surface z = 8xy and above the triangle with vertices ...

## Question:

Find the volume of the solid under the surface {eq}z = 8xy {/eq} and above the triangle with vertices {eq}(1, 1), (4, 1), \enspace and \enspace (1, 2) {/eq}

## Volume Using Double Integrals:

A double integral {eq}\displaystyle \int \int_{D} f(x, y) \; dA {/eq} is useful in finding the volume of a solid bounded above by a surface {eq}z = f(x, y) {/eq} and below by a region on the {eq}xy- {/eq}plane.

This integral can be evaluated as an iterated integral. If the limits of integration are not explicitly stated, a diagram of the region D can help you to find them.

The volume of the solid is {eq}\displaystyle\int \int_{D} f(x, y) \; dA {/eq}.

The triangle that defines D is bounded by the equations {eq}y = 1,...

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