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Use a change of variables to find the volume of the solid region lying below the surface

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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)

asked Feb 18, 2015 in CALCULUS by anonymous

2 Answers

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Step 1:

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

.

Change of variables for double integrals :

image

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 .

answered Feb 26, 2015 by joseph Apprentice
0 votes

Contd....

Step 3:

Find the Jocobian .

Definition of Jocobian :

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

image

From and .

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

image

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 image.

The volume of the solid is

image

Now use the change of variables for double integrals.

image

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 image.

image

Solution :

image.

answered Feb 26, 2015 by joseph Apprentice

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