Iterated Integral Calculator with Steps

This iterated integral calculator with steps helps you compute double integrals and other multiple integrals. It shows each step of the calculation process so you can understand how the result is derived.

What is an iterated integral?

An iterated integral is a way to evaluate multiple integrals by breaking them down into a sequence of single integrals. This process is also known as repeated integration. The most common type is a double integral, which involves integrating a function of two variables over a region in the plane.

Iterated integrals are fundamental in calculus and have applications in physics, engineering, and probability theory.

The general form of a double iterated integral is:

∫[b][a] ∫[d][c] f(x,y) dy dx

This means we first integrate with respect to y from c to d, treating x as a constant, and then integrate the result with respect to x from a to b.

How to calculate iterated integrals

Step 1: Set up the integral

First, identify the limits of integration for both variables. For a double integral, you'll need two pairs of limits: one for the inner integral and one for the outer integral.

Step 2: Integrate with respect to the inner variable

Treat the outer variable as a constant and integrate the integrand with respect to the inner variable. This will give you a new function that depends only on the outer variable.

Step 3: Integrate the result with respect to the outer variable

Now integrate the function obtained from the previous step with respect to the outer variable, using its limits of integration.

Step 4: Evaluate the final expression

The result of the second integration is the value of the double integral. This represents the volume under the surface defined by the function over the specified region.

Example

Consider the integral ∫[1][0] ∫[x²][0] (x + y) dy dx. The steps would be:

First integrate (x + y) with respect to y from 0 to x², treating x as constant.
Then integrate the result with respect to x from 0 to 1.

Example calculation

Let's calculate the iterated integral ∫[1][0] ∫[x][0] (2x + y) dy dx step by step.

Step 1: Inner integral

First, integrate (2x + y) with respect to y from 0 to x:

∫[x][0] (2x + y) dy = [2xy + (y²)/2] from 0 to x = (2x·x + x²/2) - (0 + 0) = 2x² + x²/2 = (5/2)x²

Step 2: Outer integral

Now integrate (5/2)x² with respect to x from 0 to 1:

∫[1][0] (5/2)x² dx = (5/2)[x³/3] from 0 to 1 = (5/2)(1/3 - 0) = 5/6

The value of the iterated integral is 5/6.

FAQ

What's the difference between iterated integrals and multiple integrals?: Iterated integrals are a specific method for evaluating multiple integrals by performing single integrations sequentially. Multiple integrals can also be evaluated using other methods like changing the order of integration or using polar coordinates.
When would I use an iterated integral calculator?: This calculator is useful when you need to compute integrals that depend on multiple variables, such as in physics problems involving work done over a region or probability density functions.
Can this calculator handle triple integrals?: Currently, this calculator focuses on double integrals. For triple integrals, you would need to perform three sequential integrations.
What if my integral doesn't converge?: If the integral doesn't converge, the calculator will indicate that the integral is improper or divergent. You may need to adjust your limits or consider using different integration techniques.