Calculate The Double Integral 1 1 X Y

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 in the xy-plane.

The double integral is written as ∫∫ f(x,y) dA, where dA represents an infinitesimal area element. For rectangular regions, it can be expressed as:

This represents integrating first with respect to y (from c to d) and then with respect to x (from a to b).

Key concepts

• Double integrals are used to calculate areas, volumes, and other quantities in two dimensions
• The order of integration (dy dx or dx dy) can affect the complexity of the calculation
• Double integrals are fundamental in physics for calculating work, charge, and other quantities

Calculating the double integral

To calculate ∫∫ from 0 to 1 of x y dx dy, follow these steps:

1. Identify the limits of integration: x from 0 to 1, y from 0 to 1
2. Integrate the function with respect to y first, treating x as a constant
3. Integrate the resulting expression with respect to x

Step 1: Inner integral (with respect to y)

First, integrate x y with respect to y from 0 to 1:

Step 2: Outer integral (with respect to x)

Now integrate x/2 with respect to x from 0 to 1:

The final result is 1/4.

Example calculation

Let's calculate ∫∫ from 0 to 1 of x y dx dy step by step:

1. First, integrate with respect to y:
   ∫₀¹ x y dy = x [y²/2] from 0 to 1 = x/2
2. Then integrate with respect to x:
   ∫₀¹ x/2 dx = [x²/4] from 0 to 1 = 1/4

The volume under the surface z = x y over the unit square is 1/4.

Interpretation

This result means that the average value of the function x y over the unit square is 1/4. In practical terms, this calculation could represent:

• The average density of a material over a unit area
• The average temperature distribution in a unit square region
• The average value of a product in a unit square market