Double Integral Calculator

What Is a Double Integral?

A double integral extends the idea of a single integral to functions of two variables. It's used to calculate areas, volumes, and other quantities over a two-dimensional region.

Formal Definition

A double integral of a function \( f(x, y) \) over a region \( R \) is defined as:
\( \iint_R f(x, y)\, dx\, dy \)

• f(x, y): the function being integrated
• R: the region of integration in the xy-plane
• dx dy: the infinitesimal area elements

Why It Matters

• 📐 Compute area under a surface
• 📦 Calculate the volume beneath curved surfaces
• ⚖️ Determine mass of plates with varying density
• 🌡️ Analyze heat or fluid distribution in a region

Real-Life Analogy

Imagine you’re measuring the paint needed to cover a curved surface like a warped sheet of metal. A double integral helps you slice the region into small patches, calculate the contribution from each, and sum them all up.

How to Use the Double Integral Calculator

1. Enter the function \( f(x, y) \) you want to integrate
2. Select the limits of integration for x and y
3. Choose the coordinate system (Cartesian or polar)
4. Click Calculate to compute the double integral
5. View the numerical result and step-by-step breakdown (if enabled)

Tip: Use polar coordinates for circular or radial regions to simplify integration.

Double Integral Examples

• Example 1: \( f(x, y) = x + y \) over \( 0 \leq x \leq 1, 0 \leq y \leq 2 \) → Result: 3
• Example 2: \( f(x, y) = e^{x + y} \) over square \( [0,1] \times [0,1] \) → Result: \( (e - 1)^2 \)
• Example 3: \( f(x, y) = x^2 + y^2 \) over a circle of radius 2 → Result: \( 8\pi \)

Applications of Double Integrals

• 📊 Area and volume calculations
• 🔥 Heat and mass transfer
• 🎲 Probability over 2D distributions
• ⚙️ Engineering: stress, pressure over surfaces
• 💧 Fluid flow over flat regions

Frequently Asked Questions (FAQ)