Calculadora Integral Dupla Wolfram

This calculator uses Wolfram's computational engine to solve double integrals. Double integrals are used to calculate areas, volumes, and other quantities in two-dimensional space. The calculator provides step-by-step solutions and visualizations to help you understand the process.

What is a Double Integral?

A double integral extends the concept of a single integral to two dimensions. It's used to calculate quantities like area, volume, and average values over a two-dimensional region. Double integrals are fundamental in physics, engineering, and mathematics.

The general form of a double integral is:

∫∫ f(x,y) dA = ∫[b][a] (∫[g2(x)][g1(x)] f(x,y) dy) dx

Where:

f(x,y) is the integrand function
dA represents the area element
a and b are the limits of integration for x
g1(x) and g2(x) are the lower and upper limits for y

How to Use This Calculator

Enter the integrand function in the first field (e.g., "x*y")
Specify the limits of integration for x (a and b)
Enter the lower and upper limits for y as functions of x (e.g., "0" and "x")
Click "Calculate" to get the result
Review the step-by-step solution and visualization

Note: The calculator uses Wolfram's computational engine to provide accurate results. For complex functions, the calculation may take a few seconds.

Formula and Calculation

The double integral is calculated using iterative integration:

∫∫ f(x,y) dA = ∫[b][a] (∫[g2(x)][g1(x)] f(x,y) dy) dx

The calculator performs these steps:

First integrate with respect to y from g1(x) to g2(x)
Then integrate the result with respect to x from a to b
Combine the results to get the final value

Worked Example

Let's calculate the double integral of f(x,y) = x*y over the region where 0 ≤ x ≤ 1 and 0 ≤ y ≤ x.

∫[1][0] (∫[x][0] x*y dy) dx

Step 1: First integrate with respect to y:

∫[x][0] x*y dy = x*(y²/2) evaluated from 0 to x = x*(x²/2 - 0) = x³/2

Step 2: Then integrate with respect to x:

∫[1][0] x³/2 dx = (x⁴/8) evaluated from 0 to 1 = 1/8 - 0 = 1/8

The final result is 1/8.

FAQ

What types of functions can I integrate?

The calculator can handle most common mathematical functions including polynomials, trigonometric functions, exponentials, and logarithms. For more complex functions, you may need to simplify them first.

How accurate are the results?

The calculator uses Wolfram's computational engine which provides highly accurate results. However, for very complex integrals, there may be slight numerical approximations.

Can I visualize the region of integration?

Yes, the calculator provides a visualization of the region of integration to help you understand the problem better.