Question

Use the transformation u=4x+3y,v=x+3y to evaluate the given integral for the region R bounded by the lines y=−3/4 x+1, y=− 3/4 x+2,y=− 3/1 x, and y=− 3/1 x+3 ∬ R (4x 2 +15xy+9y 2 )dxdy ∬ R (4x2 +15xy+9y 2 )dxdy=

Answer 1

To evaluate the given integral for the region R, use the transformation u=4x+3y and v=x+3y. Find the Jacobian of the transformation to convert the integral. Determine the bounds based on the transformed lines.

Step-by-step explanation:

To evaluate the given integral for the region R, we can use the transformation u=4x+3y and v=x+3y.

By finding the Jacobian of the transformation, which is 7, we can convert the integral in terms of u and v to an integral in terms of x and y.

The bounds for the new integral will be determined by the transformed lines y=-3/4x+1, y=-3/4x+2, y=-3/1x, and y=-3/1x+3.