Calculate The Iterated Integral X/y+y/x Dy Dx

The iterated integral x/y + y/x dy dx represents a double integral where we first integrate with respect to y and then with respect to x. This calculation is fundamental in multivariable calculus and has applications in physics, engineering, and other scientific fields.

What is the iterated integral x/y + y/x dy dx?

The expression x/y + y/x dy dx represents a double integral where we first integrate the integrand x/y + y/x with respect to y, and then integrate the result with respect to x. This process is called iterated integration.

Double integrals are used to calculate quantities such as area, volume, mass, and other physical properties over two-dimensional regions. The order of integration (dy dx or dx dy) can affect the result, and the limits of integration must be carefully chosen to match the region of interest.

Note: The integrand x/y + y/x is undefined at points where x=0 or y=0. Special care must be taken when setting up the integral limits to avoid these singularities.

How to calculate the iterated integral

Calculating the iterated integral x/y + y/x dy dx involves two main steps: first integrating with respect to y, then with respect to x. Here's the step-by-step process:

Identify the limits of integration for both x and y. These limits define the region over which you're integrating.
First, integrate the integrand x/y + y/x with respect to y, treating x as a constant. This gives you a function of x.
Next, integrate the result from step 2 with respect to x.
Evaluate the final expression using the given limits of integration.

The general form of the iterated integral is:

∫[a,b] ∫[c,d] (x/y + y/x) dy dx

For specific limits, you would substitute a, b, c, and d with the appropriate values. The result will be a numerical value representing the integral over the specified region.

Worked example

Let's calculate the iterated integral with limits from x=1 to x=2 and y=1 to y=2:

∫[1,2] ∫[1,2] (x/y + y/x) dy dx

First, integrate with respect to y:

∫[1,2] (x/y + y/x) dy = x ln|y| + (y²)/(2x) evaluated from y=1 to y=2

Now, evaluate the antiderivative at the limits:

[x ln(2) + (4)/(2x)] - [x ln(1) + (1)/(2x)] = x ln(2) + 2/x - 1/(2x)

Next, integrate the result with respect to x:

∫[1,2] (x ln(2) + 2/x - 1/(2x)) dx = (x²/2) ln(2) + 2 ln|x| - (1/2) ln|x| evaluated from x=1 to x=2

Finally, evaluate the antiderivative at the limits:

[ (4/2) ln(2) + 2 ln(2) - (1/2) ln(2) ] - [ (1/2) ln(2) + 2 ln(1) - (1/2) ln(1) ]

= [2 ln(2) + 2 ln(2) - 0.5 ln(2)] - [0.5 ln(2) + 0 - 0]

= (4 ln(2) - 0.5 ln(2)) - 0.5 ln(2) = 3.5 ln(2)

The final result is 3.5 ln(2), which is approximately 2.404.

FAQ

What is the difference between a single integral and a double integral?

A single integral calculates a quantity over a one-dimensional interval, while a double integral calculates a quantity over a two-dimensional region. Double integrals are used when the quantity being measured depends on two variables.

Why is the order of integration important?

The order of integration can affect the result and the complexity of the calculation. Some regions are easier to integrate in one order than another, and the limits of integration must be adjusted accordingly.

What are the applications of double integrals?

Double integrals have applications in physics, engineering, and other sciences. They can be used to calculate quantities such as area, volume, mass, and other physical properties over two-dimensional regions.