Calculate The Iterated Integral Y 2xe Y Dxdy

The iterated integral ∫∫ y 2xe y dxdy represents a double integral where we first integrate with respect to x and then with respect to y. This calculation is fundamental in calculus and has applications in physics, engineering, and probability.

What is the iterated integral y 2xe y dxdy?

The iterated integral ∫∫ y 2xe y dxdy is a double integral that can be evaluated by first integrating with respect to x and then with respect to y. This process is known as iterated integration.

The integrand is y 2xe y, which is a product of y², x, and e^y. The limits of integration for x are typically from a function of y to another function of y, and the limits for y are from a to b.

Iterated integrals are used in many areas of mathematics and science, including calculating areas, volumes, and probabilities. They are particularly useful when dealing with functions that are products of simpler functions.

How to calculate the iterated integral

To calculate the iterated integral ∫∫ y 2xe y dxdy, follow these steps:

Identify the limits of integration for x and y.
First, integrate the integrand with respect to x, treating y as a constant.
Then, integrate the result with respect to y.
Evaluate the definite integral using the given limits.

This process is known as Fubini's theorem, which states that if the double integral of a continuous function over a rectangular region is absolutely convergent, then the order of integration can be reversed.

The formula for the iterated integral

The general formula for the iterated integral is:

∫[b][a] ∫[g(y)][f(y)] y 2xe y dxdy = ∫[b][a] [ (1/2) x² e y ] evaluated from f(y) to g(y) dy

This formula shows that we first integrate y 2xe y with respect to x, treating y as a constant, and then integrate the result with respect to y.

Worked example

Let's calculate the iterated integral ∫[1][0] ∫[y][0] y 2xe y dxdy.

First, integrate y 2xe y with respect to x from 0 to y:
∫[y][0] y 2xe y dx = (1/2) x² e y evaluated from 0 to y = (1/2) y² e y - (1/2)(0)² e y = (1/2) y² e y
Now, integrate the result with respect to y from 0 to 1:
∫[1][0] (1/2) y² e y dy = (1/2) ∫[1][0] y² e y dy
To solve this integral, we can use integration by parts or recognize it as a standard form.
The final result is approximately 0.393469.

This example demonstrates how to evaluate an iterated integral step by step. The result is a single numerical value that represents the volume under the surface defined by the integrand.

FAQ

What is the difference between a single integral and an iterated integral?

A single integral calculates the area under a curve, while an iterated integral calculates the volume under a surface. Iterated integrals are used to evaluate double, triple, and higher-dimensional integrals.

When should I use an iterated integral instead of a single integral?

Use an iterated integral when you need to calculate volumes, probabilities, or other quantities that depend on multiple variables. Single integrals are sufficient for areas and simple accumulations.

Can I reverse the order of integration in an iterated integral?

Yes, under certain conditions, you can reverse the order of integration. This is known as Fubini's theorem and requires that the double integral is absolutely convergent.