Calculate The Iterated Integral X Y 2

This guide explains how to calculate the iterated integral of x y 2, including the step-by-step process, common pitfalls, and practical applications. The interactive calculator on this page makes it easy to compute the result for any given limits.

What is an iterated integral?

An iterated integral is a double integral that is evaluated by integrating with respect to one variable first, then the other. For the function f(x, y) = x y 2, we first integrate with respect to y, then with respect to x.

Iterated integrals are used in physics, engineering, and economics to calculate quantities like work, probability, and expected values.

How to calculate the iterated integral x y 2

The process involves:

Identify the limits of integration for x and y
Integrate with respect to y first, treating x as a constant
Integrate the result with respect to x

The general formula for the iterated integral of x y 2 is:

∫[a to b] ∫[c to d] x y 2 dy dx = ∫[a to b] [ (x y 3)/3 ] evaluated from c to d dx

Worked example

Let's calculate the iterated integral of x y 2 from x=0 to x=2 and y=0 to y=3.

Step 1: First integration (with respect to y)

∫[0 to 3] x y 2 dy = x ∫[0 to 3] y 2 dy = x [ (y 3)/3 ] from 0 to 3 = x (27/3 - 0) = 9x

Step 2: Second integration (with respect to x)

∫[0 to 2] 9x dx = 9 [ (x 2)/2 ] from 0 to 2 = 9 (2 - 0) = 18

The final result is 18.

FAQ

What's the difference between iterated and double integrals?: Iterated integrals are evaluated by integrating with respect to one variable at a time, while double integrals are evaluated simultaneously over a region.
When should I use an iterated integral?: Use iterated integrals when the limits of integration are rectangular (constant with respect to one variable) or when the function is easier to integrate in one order.
Can I change the order of integration?: Yes, but you must adjust the limits of integration accordingly. The result will be the same if the region of integration is simple and connected.