Question

Find the volume of the given solid.

 Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)

Answer 1

Step 1:

The surface is and the vertices of the triangle are .

Volume of the solid :

The volume of the solid V under the surface and lies above the region

then .

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

(3) Connect the plotted vertices to a smooth triangle.

Observe the graph :

The limits of y are varying from 1 to 2 , so .

Find the bounds for x :

Consider the points .

From the points, coordinates are equal then the equation of the line parallel to axis.

So the equation of the line is

Consider the points .

Using two points form of a line equation is .

Substitute in the line equation.

Therefore then .

Step 2:

Find the volume of the solid.

The obtained region is .

Then

Solution :

The volume of the solid is .