   Chapter 15.2, Problem 25E

Chapter
Section
Textbook Problem

Find the volume of the given solid.25. Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)

To determine

To find: The volume of the solid that lies under the surface and above the triangle.

Explanation

Given:

The surface is z=xy .

The vertices of triangle is, (1,1),(4,1),(1,2) .

Formula used:

The volume of the solid, V=DzdA , where, z is the given function.

Calculation:

The equation of the side of the triangle is x=73y and from the vertices of the triangle it is observed that y varies from 1 and 2. So, the volume of the solid is computed as follows.

V=RzdA=12173y(xy)dxdy

First, compute the integral with respect to x.

V=12[x2y2]173ydy

Apply the limit value for x,

V=12[((73y)2y2)((1)2y2)]dy=1212[49y+9y342y2y]dy=1212[48y+9y342y2]dy

Compute the integral with respect to y

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