Question

Use a change of variables to find the volume of the solid region lying below the surface z = f (x, y) and above the plane region R.

R: region bounded by the square with vertices (4, 0), (6, 2), (4, 4), (2, 2)

Answer 1

Step 1:

The function is and region bounded by the square with vertices are

.

Change of variables for double integrals :

First find the change of variables using the vertices .

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

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

Step 2:

Observe the graph, Consider the vertices .

Using two points form of a line equation is .

Substitute in the line equation.

Observe the graph, Consider the vertices .

Using two points form of a line equation is .

Substitute in the line equation.

Observe the graph, Consider the vertices .

Using two points form of a line equation is .

Substitute in the line equation.

Observe the graph, Consider the vertices .

Using two points form of a line equation is .

Substitute in the line equation.

The obtained line equations are

From above equations, consider .

Then and .

Answer 2

Contd....

Step 3:

Find the Jocobian .

Definition of Jocobian :

If , then the Jocobian for x and y with respect to u and v is

From and .

The partial derivatives of x and y with respect to u and v are

Step 4:

Find the volume of the solid.

Volume of the solid :

The volume of the solid V under the surface and lies above the region R, using the change

of variables then .

The volume of the solid is

Now use the change of variables for double integrals.

Find the bounds for S in the plane using the bounds for R in the plane.

The region bounded by the S in the plane is .

Solution :

.