Evaluate the integral by making an appropriate change of variables.
25. () dA, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)
To evaluate: The integral by change of variable method.
The integral is where the trapezoidal region with vertices and .
Property used: Change of Variable
Change of Variable in double integral is given by, (1)
Definition used: Jacobian transformation
From the given integral observe and let and .
Obtain the value of .
Find the partial derivative of x and y with respect to u and v. then and and then and .
Substitute the corresponding values in equation (2),
This is in the form of inverse of Jacobian transformation but obtain Jacobian transformation . Therefore the value of Jacobian 2 will be
Substitute the given vertices of trapezoidal in and to get the image of the trapezoidal with vertices that is and
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