Evaluate Iterated Integral Calculator

An iterated integral is a mathematical operation that involves integrating a function with respect to one variable, then integrating the result with respect to another variable. This process can be repeated for multiple variables, making iterated integrals fundamental in calculus and physics.

What is an iterated integral?

An iterated integral is a sequence of integrals performed one after another. For a function of two variables, f(x, y), the double integral is written as:

∫∫ f(x, y) dA = ∫[b to a] (∫[d to c] f(x, y) dy) dx

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

The order of integration matters. For some functions, changing the order of integration can lead to different results. This is known as the Fubini's theorem.

How to evaluate an iterated integral

Step 1: Determine the order of integration

Choose the order of integration based on the region of integration. For rectangular regions, either order is acceptable. For more complex regions, you may need to sketch the region to determine the correct order.

Step 2: Integrate with respect to the inner variable

First, integrate the function with respect to the inner variable (usually y). Treat the outer variable (x) as a constant during this integration.

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

Now, integrate the result from step 2 with respect to the outer variable (x). This will give you the final value of the iterated integral.

Remember that the limits of integration may change based on the order of integration. Always double-check your limits when changing the order of integration.

Using the calculator

Our iterated integral calculator makes evaluating these integrals quick and easy. Simply enter your function, specify the limits of integration, and choose the order of integration. The calculator will handle the rest.

Example calculation

Let's evaluate the integral of x²y from x=0 to x=2 and y=0 to y=3:

∫[2 to 0] (∫[3 to 0] x²y dy) dx

First, we integrate with respect to y:

∫[3 to 0] x²y dy = x²(3²/2 - 0²/2) = 4.5x²

Then we integrate with respect to x:

∫[2 to 0] 4.5x² dx = 4.5(2³/3 - 0³/3) = 4.5 × 2.666... ≈ 12

The calculator will give you the exact value of 12 for this integral.

Common applications

Iterated integrals are used in various fields including:

Physics for calculating work done by variable forces
Engineering for finding centroids and moments of inertia
Probability for calculating joint probabilities
Computer graphics for rendering 3D objects

Common iterated integral applications
Field	Application
Physics	Calculating work done by variable forces
Engineering	Finding centroids and moments of inertia
Probability	Calculating joint probabilities
Computer Graphics	Rendering 3D objects

FAQ

What's the difference between iterated integrals and multiple integrals?

Iterated integrals are a specific method for evaluating multiple integrals by performing single integrals sequentially. Multiple integrals can also be evaluated using other methods like changing the order of integration or using polar coordinates.

When should I change the order of integration?

You should change the order of integration when the limits become simpler or when the region of integration is easier to describe in the new order. This often happens with non-rectangular regions.

What happens if I change the order of integration?

The value of the integral remains the same if the function is continuous and the region of integration is simple. However, the limits of integration will change, and the intermediate results may differ.