Area of Double Integral Calculator

What is a Double Integral?

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

Double integrals have applications in physics (calculating mass distributions), engineering (determining surface areas), and probability (calculating expected values over regions).

How to Calculate the Area of a Double Integral

To compute the area using a double integral:

1. Define the function z = f(x,y) that represents the surface
2. Determine the region D over which to integrate
3. Set up the double integral ∫∫_D f(x,y) dA
4. Evaluate the integral using appropriate techniques (iterated integrals, polar coordinates, etc.)

The Formula

The area A of a surface defined by z = f(x,y) over region D is given by:

For rectangular regions, this becomes:

Where:

• f(x,y) is the function defining the surface
• D is the region of integration
• dA is the differential area element

Worked Example

Let's calculate the area under the surface z = x² + y² over the square region [0,1] × [0,1].

1. Set up the double integral:
   A = ∫₀¹ ∫₀¹ (x² + y²) dy dx
2. Integrate with respect to y first:
   ∫₀¹ (x² + y²) dy = [x²y + (y³)/3]₀¹ = x² + 1/3
3. Now integrate with respect to x:
   ∫₀¹ (x² + 1/3) dx = [(x³)/3 + (x)/3]₀¹ = 1/3 + 1/3 = 2/3

The area under the surface is 2/3 square units.