Calculate The Double Integral Xe X 2

Double integrals are used to calculate quantities that depend on two variables, such as area, volume, mass, and probability. This guide explains how to calculate the double integral of xe x 2, including the formula, step-by-step calculation, and practical applications.

What is a Double Integral?

A double integral extends the concept of a single integral to two dimensions. It calculates the volume under a surface defined by a function of two variables, z = f(x, y), over a region in the xy-plane.

The double integral is expressed as:

∫∫_R f(x,y) dA = ∫_{a}^{b} ∫_{c(x)}^{d(x)} f(x,y) dy dx

Where R is the region of integration, and dA represents an infinitesimal area element.

Formula for xe x 2

The double integral of xe x 2 can be calculated using the following formula:

∫∫_R xe x 2 dA = ∫_{a}^{b} ∫_{c(x)}^{d(x)} xe x 2 dy dx

This integral represents the volume under the surface xe x 2 over the specified region R.

How to Calculate

To calculate the double integral of xe x 2:

Define the region of integration R in the xy-plane.
Set up the iterated integral with appropriate limits.
Integrate with respect to y first, then with respect to x.
Evaluate the definite integrals using the Fundamental Theorem of Calculus.

For complex regions, it may be necessary to use polar coordinates or other coordinate transformations.

Worked Example

Let's calculate the double integral of xe x 2 over the region R defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ x.

∫_{0}^{1} ∫_{0}^{x} xe x 2 dy dx

First, integrate with respect to y:

∫_{0}^{x} xe x 2 dy = xe x 2 ∫_{0}^{x} dy = xe x 2 [y]_{0}^{x} = xe x 2 * x = xe x 3

Then integrate with respect to x:

∫_{0}^{1} xe x 3 dx = ∫_{0}^{1} e x 3 dx = [e x 4 / 4]_{0}^{1} = (e - 1)/4

The final result is (e - 1)/4.

Applications

Double integrals are used in various fields including:

Physics: Calculating mass distributions
Engineering: Determining moments of inertia
Probability: Calculating joint probabilities
Economics: Analyzing production functions

FAQ

What is the difference between single and double integrals?: A single integral calculates quantities that depend on one variable, while a double integral extends this to two variables.
When would I use polar coordinates for a double integral?: Polar coordinates are often used when the region of integration is circular or has radial symmetry.
Can double integrals be negative?: Yes, if the function being integrated is negative over part of the region, the double integral can be negative.
What are some common applications of double integrals?: Common applications include calculating areas, volumes, masses, and probabilities in two dimensions.
How do I handle double integrals with complex regions?: For complex regions, it may be necessary to break the integral into simpler parts or use coordinate transformations.