Double Integral Over Rectangular Region Calculator

What is a Double Integral Over a Rectangular Region?

A double integral extends the concept of single integration to two dimensions. For a rectangular region, we integrate a function f(x,y) first with respect to y over the interval [c,d], then integrate the resulting function with respect to x over [a,b].

This process calculates the volume under the surface z = f(x,y) and above the rectangular region in the xy-plane. The result represents the total accumulation of the function over the given area.

The Formula

The double integral is evaluated by first integrating with respect to y from c to d, then integrating the result with respect to x from a to b. For simple functions, this can be done analytically, while more complex functions may require numerical methods.

How to Calculate a Double Integral Over a Rectangular Region

1. Identify the function f(x,y) to be integrated
2. Determine the rectangular region bounds: x from a to b and y from c to d
3. Integrate f(x,y) with respect to y from c to d, treating x as a constant
4. Integrate the result from step 3 with respect to x from a to b
5. Evaluate the final expression to get the double integral value

For functions that can't be integrated analytically, numerical methods like the trapezoidal rule or Simpson's rule can approximate the integral.

Worked Example

Calculate ∫[0,2] ∫[1,3] (x² + y) dy dx

1. First integrate with respect to y: ∫[1,3] (x² + y) dy = [x²y + y²/2] from 1 to 3 = (x²*3 + 9/2) - (x²*1 + 1/2) = 2x² + 4
2. Now integrate the result with respect to x: ∫[0,2] (2x² + 4) dx = [2x³/3 + 4x] from 0 to 2 = (16/3 + 8) - (0 + 0) = 16/3 + 24/3 = 40/3 ≈ 13.333

The exact value of this double integral is 40/3.