Step by Step Double Integral Calculator

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 over a region in the xy-plane, or the area of a region when integrated over a constant function.

Double integrals are expressed as:

Where R is the region of integration, f(x,y) is the integrand, and dA represents the area element.

How to Calculate a Double Integral

Calculating a double integral involves several steps:

1. Define the region of integration R
2. Determine the limits of integration (a, b, g1(x), g2(x))
3. Set up the iterated integral
4. Integrate with respect to the inner variable first
5. Integrate the result with respect to the outer variable

Step-by-Step Calculation Method

Step 1: Define the Region

First, sketch the region R in the xy-plane and determine its boundaries. For example, a rectangle from (0,0) to (2,3).

Step 2: Set Up the Integral

For the rectangle example, the double integral becomes:

Step 3: Integrate with Respect to y

First, integrate the inner integral with respect to y:

Step 4: Integrate with Respect to x

Then integrate the result with respect to x:

Example Calculation

Let's compute ∫∫_R (x² + y) dA over the rectangle [0,2]×[0,3].

1. Set up the integral: ∫_{0}^{2} ∫_{0}^{3} (x² + y) dy dx
2. Integrate inner integral: ∫ (x² + y) dy = x²y + y²/2 evaluated from 0 to 3
3. Result: (3x² + 9/2) - (0 + 0) = 3x² + 4.5
4. Integrate outer integral: ∫ (3x² + 4.5) dx = x³ + 4.5x evaluated from 0 to 2
5. Final result: (8 + 9) - (0 + 0) = 17

Common Applications

Double integrals are used in various fields including:

• Calculating areas and volumes
• Finding centers of mass and moments of inertia
• Computing probabilities in probability theory
• Solving partial differential equations