Wolfram Alpha Definite Double Integral Calculator

What is a Definite Double Integral?

A definite double integral extends the concept of single integrals to two dimensions. It calculates the volume under a surface z = f(x,y) bounded by curves in the xy-plane. The integral is written as:

Where:

• f(x,y) is the integrand function
• R is the region of integration
• a and b are the x-axis limits
• u(x) and v(x) are the y-axis limits as functions of x

Double integrals have applications in physics, engineering, and probability, particularly for calculating masses, moments, and probabilities over two-dimensional regions.

How to Use This Calculator

1. Enter your integrand function in the first field (e.g., "x*y" or "sin(x+y)")
2. Specify the x-axis limits (a and b)
3. Enter the y-axis limits as functions of x (u(x) and v(x))
4. Click "Calculate" to get the result
5. View the visualization of the region and function

The Formula Explained

The definite double integral is calculated using iterative integration:

The process involves:

1. First integrating with respect to y from u(x) to v(x)
2. Then integrating the result with respect to x from a to b

The calculator performs these steps automatically using Wolfram Alpha's computational engine.

Worked Example

Calculate the volume under z = x² + y² bounded by x from 0 to 1 and y from 0 to x.

The calculator would:

1. First integrate (x² + y²) with respect to y from 0 to x
2. Then integrate the result with respect to x from 0 to 1
3. Return the final volume value

For this specific example, the exact volume is 5/24.