Use the given transformation to evaluate the given integral, where r is the triangular region with vertices (0, 0), (8, 1), and (1, 8). (x − 3y) da r , x = 8u + v, y = u + 8v

We first obtain the equation of the lines bounding R.

For the line with points (0, 0) and (8, 1), the equation is given by:



For the line with points (0, 0) and (1, 8), the equation is given by:



For the line with points (8, 1) and (1, 8), the equation is given by:



The Jacobian determinant is given by



The integrand x - 3y is transformed as 8u + v - 3(u + 8v) = 8u + v - 3u - 24v = 5u - 23v

Therefore, the integration is given by: