Question

Evaluate the integral ∬_R (x/y - y/x) dA over the rectangular region, R = 1 ≤ x ≤ 2, 1 ≤ y ≤ 5.

Answer 1

To evaluate the given integral, we need to calculate the double integral over the rectangular region R = 1 ≤ x ≤ 2, 1 ≤ y ≤ 5. The integrand is (x/y - y/x). We can rewrite this as (x^2/x*y - y/x). Now, we can evaluate the double integral by integrating with respect to x first and then with respect to y.

Step-by-step explanation:

To evaluate the given integral, we need to calculate the double integral over the rectangular region R = 1 ≤ x ≤ 2, 1 ≤ y ≤ 5. The integrand is (x/y - y/x). We can rewrite this as (x^2/x*y - y/x). Now, we can evaluate the double integral by integrating with respect to x first and then with respect to y.

1. Integrate with respect to x: ∫(1 ≤ x ≤ 2) (x^2/x*y - y/x)dx = ∫(1 ≤ x ≤ 2) (x/y - y/x)dx = [x^2/2y - yln|x|] from x = 1 to x = 2

2. Integrate with respect to y: ∫(1 ≤ y ≤ 5) ([x^2/2y - yln|x|] from x = 1 to x = 2)dy = ∫(1 ≤ y ≤ 5) [2/y - yln(2) + yln(1)]dy = [2ln|y| - y^2ln(2) + y] from y = 1 to y = 5

Substituting the limits gives the final answer:

The value of the given double integral is [2ln(5) - 25ln(2) + 5].