Final answer:
To calculate the iterated integral of (x/y + y/x) dydx, each part is integrated with respect to y, evaluated between 1 and 4, and then integrated with respect to x between 1 and 16. The iterated integral adds the areas under the curves defined by x/y and y/x over the given limits.
Step-by-step explanation:
To calculate the iterated integral ∫¹⁶₁ ∫⁴₁ (x/y + y/x) dydx, you first integrate with respect to y, while treating x as a constant, and then integrate the result with respect to x. The given integral represents the sum of two separate integrals due to the properties of integrals over sums.
First, consider the integral of x/y with respect to y:
∫ (x/y) dy = x ∫ (1/y) dy = x ln|y|
And for the integral of y/x with respect to y:
∫ (y/x) dy = (1/x) ∫ y dy = ½ (y^2/x)
Now, putting these together, we have:
∫ (∫ (x/y + y/x) dy) dx = ∫ (x ln|y| + ½(y^2/x)) dydx
You then evaluate these from y=1 to y=4, and substitute these values into the expression before finally integrating with respect to x from x=1 to x=16.
Performing these steps provides the value of the iterated integral, which represents the summation of the area under the curve defined by the functions within the given limits.