Use the given transformation to evaluate the integral.
double integral R (x − 8y) dA,
where R is the triangular region with vertices (0, 0), (7, 1), and (1, 7).
x = 7u + v, y = u + 7v
To evaluate the given integral using the given transformation, we need to determine the new limits of integration in terms of the new variables u and v.
The given transformation relates the original variables (x, y) to the new variables (u, v) as follows:
x = 7u + v
y = u + 7v
To determine the new limits of integration, we first need to find the equations of the boundary lines of the triangular region R in terms of the new variables.
The vertices of the triangular region R are (0, 0), (7, 1), and (1, 7).
For the first vertex (0, 0), we substitute the values of x and y into the transformation equations:
0 = 7u + v
0 = u + 7v
Solving these equations simultaneously, we find that u = 0 and v = 0.
For the second vertex (7, 1), we substitute the values of x and y into the transformation equations:
7 = 7u + v
1 = u + 7v
Solving these equations simultaneously, we find that u = 0 and v = 1/7.
For the third vertex (1, 7), we substitute the values of x and y into the transformation equations:
1 = 7u + v
7 = u + 7v
Solving these equations simultaneously, we find that u = 1/7 and v = 1.
So, the new limits of integration for u are from 0 to 1/7, and the new limits of integration for v are from 0 to 1.
Now, we can rewrite the integral in terms of the new variables u and v:
double integral R (x − 8y) dA = double integral R [(7u + v) − 8(u + 7v)] |J| du dv,
where |J| is the determinant of the Jacobian matrix of the transformation:
|J| = | (∂x/∂u) (∂x/∂v) |
| (∂y/∂u) (∂y/∂v) |
= |7 1 |
|1 7 |
= 48.
Therefore, the integral becomes:
double integral R [(7u + v) − 8(u + 7v)] |J| du dv
= 48 double integral R [7u + v − 8u − 56v] du dv
= 48 double integral R (− u − 55v) du dv.
Now, you can evaluate the integral by integrating the expression (− u − 55v) with respect to u first, using the new limits of integration for u (0 to 1/7). After that, you can integrate the resulting expression with respect to v, using the limits (0 to 1).