Evaluate double integral ln((x-y)/(x+y)) dy dx where the double integral region is the triangle with vertices (1,0),(4,3), (4,1). Hint: use a transformation with the Jacobian.

This type of problem is discussed on pages 3-5 at

http://math.etsu.edu/multicalc/prealpha/Chap4/Chap4-4/printversion.pdf

nope

To evaluate the double integral using the given hint, we will perform a change of variables using a transformation with the Jacobian.

Let's define a new variable u = x - y and v = x + y. We can rewrite these equations in terms of x and y as follows:

x = (u + v)/2 (Eq. 1)
y = (v - u)/2 (Eq. 2)

Now, we need to find the Jacobian of this transformation. The Jacobian matrix J is given by:

J = | ∂x/∂u ∂x/∂v |
| ∂y/∂u ∂y/∂v |

To find the partial derivatives, we differentiate Eq. 1 and Eq. 2 with respect to u and v:

∂x/∂u = 1/2 (Eq. 3)
∂x/∂v = 1/2 (Eq. 4)
∂y/∂u = -1/2 (Eq. 5)
∂y/∂v = 1/2 (Eq. 6)

Substituting these values into the Jacobian matrix, we have:

J = | 1/2 1/2 |
| -1/2 1/2 |

Now, let's compute the determinant of J:

det(J) = (1/2)(1/2) - (-1/2)(1/2)
= 1/4 + 1/4
= 1/2

The absolute value of the determinant represents the scaling factor of the area element, which is 1/2. This means that dxdy = (1/2)dudv.

Next, we need to find the region of integration in the uv-plane that corresponds to the triangle with vertices (1,0), (4,3), and (4,1) in the xy-plane.

The transformation equations from xy-plane to uv-plane are:

u = x - y
v = x + y

Let's determine the new coordinates in the uv-plane for the given vertices:

(1,0) --> u = 1 - 0 = 1, v = 1 + 0 = 1
(4,3) --> u = 4 - 3 = 1, v = 4 + 3 = 7
(4,1) --> u = 4 - 1 = 3, v = 4 + 1 = 5

The region of integration in the uv-plane is a triangle with vertices (1,1), (1,7), and (3,5).

Now, we can rewrite the double integral with the new variables:

∫∫[ln((x-y)/(x+y))] dy dx = (1/2)∫∫[ln(u/v)] dv du

The limits of integration for the new variables u and v can be determined from the vertices of the triangle in the uv-plane:

For u, the limits are from u = 1 to u = 3.
For v, the limits vary depending on the line segments forming the triangle:
- For v at the left side of the triangle, the lower limit is a line connecting (1,1) and (3,5):
v = 1 + (7 - 1)(u - 1)/(3 - 1)
v = 1 + 3(u - 1)/2
- For v at the right side of the triangle, the upper limit is a line connecting (1,7) and (3,5):
v = 7 - (7 - 5)(u - 1)/(3 - 1)
v = 7 - 2(u - 1)/2

Thus, the limits of integration for v are:
- For the left side of the triangle, v varies from the lower limit function to 7.
- For the right side of the triangle, v varies from 5 to the upper limit function.

Finally, we can evaluate the integral:

∫∫[ln(u/v)] dv du = (1/2)∫[ln(u/v)] from v = 1 + 3(u - 1)/2 to v = 7 du
+ (1/2)∫[ln(u/v)] from v = 5 to v = 7 - 2(u - 1)/2 du

This completes the setup for evaluating the double integral using a transformation with the Jacobian. The next step is to perform the integration using the given limits and the properties of natural logarithms.