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Integral Doble Calculadora

Reviewed by Calculator Editorial Team

A double integral calculator is a powerful tool for solving two-dimensional integrals in calculus. This calculator helps you compute the volume under a surface, the area of a region, or other applications of double integrals in mathematics and physics.

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 z = f(x,y) over a region D in the xy-plane. Double integrals have applications in physics, engineering, and probability.

Double Integral Formula

∫∫D f(x,y) dA = ∫ab ∫u(x)v(x) f(x,y) dy dx

Where D is the region of integration, and dA is the area element.

Types of Double Integrals

There are two main types of double integrals:

  1. Iterated Integrals: Solved by integrating first with respect to one variable and then the other.
  2. Green's Theorem: Relates a double integral over a region to a line integral around its boundary.

How to Use This Calculator

This double integral calculator provides a user-friendly interface to solve double integrals. Follow these steps:

  1. Enter the function you want to integrate in the function field.
  2. Specify the limits of integration for both x and y.
  3. Click "Calculate" to compute the integral.
  4. Review the result and chart visualization if available.

Note

This calculator uses numerical methods for approximation. For exact results, symbolic computation software may be needed.

Formula and Calculation

The double integral is calculated using the iterated integral method. The calculator performs the following steps:

  1. Evaluate the inner integral with respect to y.
  2. Integrate the result with respect to x over the specified limits.
  3. Return the final result and visualize the function if possible.

Calculation Steps

1. Compute ∫u(x)v(x) f(x,y) dy

2. Then compute ∫ab [result from step 1] dx

Worked Example

Let's compute the double integral of f(x,y) = x²y over the region D defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ x.

Example Calculation

∫∫D x²y dA = ∫01 ∫0x x²y dy dx

First, compute the inner integral: ∫0x x²y dy = (x²/2)y² evaluated from 0 to x = x⁴/2

Then compute the outer integral: ∫01 x⁴/2 dx = (x⁵/10) evaluated from 0 to 1 = 1/10

The result of this double integral is 0.1.

Frequently Asked Questions

What is the difference between single and double integrals?
A single integral calculates area under a curve, while a double integral calculates volume under a surface or area of a region in two dimensions.
When would I use a double integral calculator?
Use this calculator when you need to compute volumes, areas, or other quantities involving two-dimensional integrals in calculus or physics.
Can this calculator handle polar coordinates?
Currently, this calculator works with Cartesian coordinates. For polar coordinates, you may need to convert to Cartesian first.
Is the result exact or an approximation?
The calculator provides an approximate result using numerical methods. For exact results, symbolic computation software is recommended.
What if my function is complex?
The calculator can handle most standard mathematical functions. For very complex functions, you may need to simplify the expression first.