1. What is a Double Integral?

A double integral is an integral over a two-dimensional area that calculates the volume under a surface defined by a function f(x,y) over a region in the xy-plane.

2. How Does the Calculator Work?

The calculator computes the double integral:

Where:

• \( f(x,y) \) — The function to integrate
• \( R \) — The rectangular region [a,b]×[c,d]
• \( dA \) — The differential area element (dx dy or dy dx)

Explanation: The calculator performs iterated integration, first integrating with respect to one variable while treating the other as constant, then integrating the result with respect to the second variable.

3. Applications of Double Integrals

Details: Double integrals are used to calculate volumes, average values, center of mass, moment of inertia, and probability distributions in two dimensions.

4. Using the Calculator

Tips: Enter the function f(x,y) using standard mathematical notation. Specify the rectangular region of integration with lower and upper limits for both x and y.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be integrated?
A: The calculator can handle polynomial, trigonometric, exponential, and logarithmic functions over rectangular domains.

Q2: How are non-rectangular regions handled?
A: For non-rectangular regions, the limits of integration must be expressed as functions of the other variable.

Q3: What numerical methods are used?
A: The calculator typically uses adaptive quadrature methods for accurate numerical integration.

Q4: Are there limitations to this calculator?
A: The calculator may have difficulty with improper integrals or functions with discontinuities in the region of integration.

Q5: Can I change the order of integration?
A: Yes, according to Fubini's Theorem, the order can be changed for continuous functions over rectangular regions.