Calculated The Iterated Integral X Y Y X Dxdy

This guide explains how to calculate the double integral ∫∫ x/y from y to x dxdy. We'll cover the mathematical approach, provide a step-by-step calculation method, and include an interactive calculator to compute the result for specific values.

What is the iterated integral x/y from y to x dxdy?

The iterated integral ∫∫ x/y from y to x dxdy represents a double integral where we first integrate with respect to x from y to x, and then integrate the result with respect to y. This type of integral is commonly encountered in calculus and physics problems involving area calculations under curves.

The integral can be written as:

∫[a to b] (∫[y to x] (x/y) dx) dy

Where a and b are the limits of integration for y, and x is the upper limit for the inner integral with respect to x.

How to calculate this integral

Calculating the iterated integral ∫∫ x/y from y to x dxdy involves two main steps:

First, compute the inner integral with respect to x from y to x.
Then, integrate the result with respect to y from the lower limit a to the upper limit b.

Step 1: Compute the inner integral

The inner integral is ∫[y to x] (x/y) dx. Since x is a constant with respect to x, we can factor it out:

∫[y to x] (x/y) dx = x ∫[y to x] (1/y) dx

The integral of 1/y is ln|y|, so:

x [ln|x| - ln|y|] = x (ln x - ln y)

Step 2: Compute the outer integral

Now we integrate the result from step 1 with respect to y from a to b:

∫[a to b] x (ln x - ln y) dy

This can be split into two integrals:

x ln x ∫[a to b] dy - x ∫[a to b] ln y dy

The first integral is straightforward:

x ln x (b - a)

The second integral requires integration by parts. Let u = ln y and dv = dy:

∫ ln y dy = y ln y - ∫ y (1/y) dy = y ln y - y

Evaluating from a to b:

[b ln b - b] - [a ln a - a] = (b ln b - a ln a) - (b - a)

Putting it all together, the final result is:

x (b - a) ln x - x [(b ln b - a ln a) - (b - a)]

Worked example

Let's calculate the integral with a = 1, b = 2, and x = 3:

Example Calculation

Given a = 1, b = 2, x = 3:

∫[1 to 2] (∫[y to 3] (x/y) dx) dy

First compute the inner integral:

∫[y to 3] (3/y) dx = 3 (ln 3 - ln y)

Now compute the outer integral:

∫[1 to 2] 3 (ln 3 - ln y) dy = 3 ln 3 (2 - 1) - 3 [(2 ln 2 - 1 ln 1) - (2 - 1)]

= 3 ln 3 (1) - 3 [(2 ln 2 - 0) - 1] = 3 ln 3 - 3 (2 ln 2 - 1)

= 3 ln 3 - 6 ln 2 + 3 ≈ 3(1.0986) - 6(0.6931) + 3 ≈ 3.2958 - 4.1586 + 3 ≈ 2.1372

FAQ

What is the difference between single and double integrals?

A single integral calculates the area under a curve for a function of one variable. A double integral extends this concept to two variables, calculating volume under a surface or area in a plane with two variables.

When would I use an iterated integral?

Iterated integrals are used in physics for calculating work, in probability for joint distributions, and in engineering for analyzing surfaces and volumes. They're particularly useful when the limits of integration are not straightforward.

How do I know which variable to integrate first?

The order of integration depends on the problem. For iterated integrals, you typically integrate with respect to the variable that has simpler limits first. In this case, we integrate with respect to x first because its limits are constants.