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 of two variables. It extends the concept of a single integral to higher dimensions.

2. How Does the Calculator Work?

The calculator computes the double integral:

Where:

• \( f(x,y) \) — The function to integrate
• \( [a,b] \) — Integration limits for x
• \( [c,d] \) — Integration limits for y

Explanation: The double integral calculates the signed volume between the surface z = f(x,y) and the xy-plane over the rectangular region [a,b]×[c,d].

3. Importance of Double Integrals

Details: Double integrals are fundamental in physics for calculating mass, center of mass, moments of inertia, and in probability for joint probability distributions.

4. Using the Calculator

Tips: Enter a valid mathematical function of x and y, and the integration limits. The function should be continuous over the integration region for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be integrated?
A: The calculator can handle polynomial, trigonometric, exponential, and other standard mathematical functions.

Q2: Can I integrate over non-rectangular regions?
A: This calculator currently supports rectangular regions only. For more complex regions, you would need to express the limits as functions.

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

Q4: How precise are the results?
A: Results are typically accurate to several decimal places, depending on the function's behavior.

Q5: Can I use variables other than x and y?
A: The calculator expects variables x and y. For other variables, you would need to substitute accordingly.