Free Online
Double Integral Calculator
Solve iterated integrals for volume and area calculations. Accurate, fast, and easy to use.
Double Integral Calculator
Enter your function using the keypad or keyboard.
Result will appear here
Why Use Our Double Integral Solver?
Double integrals can be complex. We simplify the process by handling the integration steps for you.
Iterated Integration
Solve the inner integral first, then use the result for the outer integral.
High Precision
Get exact results for definite and indefinite double integrals.
Volume & Area
Perfect for calculating volumes under surfaces and areas of regions.
What is a Double Integral?
A double integral is an integral where the integrand is a function of two variables, typically f(x, y), and the integration is performed over a region in the xy-plane. It is denoted by two integral signs: ∫∫.
Double integrals are used to calculate the volume under a surface z = f(x, y), the area of a region, the mass of a lamina with variable density, and other physical quantities like center of mass and moments of inertia.
How to Solve Double Integrals
Double integrals are typically solved as iterated integrals. This means you perform two single integrations in sequence:
- Inner Integral: Integrate with respect to the first variable (e.g., dy), treating the other variable (x) as a constant.
- Outer Integral: Integrate the result of the first step with respect to the second variable (e.g., dx).
How to Use This Calculator for Double Integrals
Since this calculator solves single integrals, you can solve a double integral by performing the steps iteratively:
- 1Enter the inner function f(x, y). Treat the outer variable as a constant (e.g., if integrating with respect to y, treat x as a number).
- 2Calculate the integral with respect to the inner variable.
- 3Take the result and integrate it with respect to the outer variable.
Frequently Asked Questions
What is the difference between a double integral and a single integral?
A single integral finds the area under a curve, while a double integral finds the volume under a surface over a 2D region.
Can I change the order of integration?
Yes, Fubini's Theorem states that for continuous functions, the order of integration (dx dy vs dy dx) does not change the result, provided the limits are adjusted accordingly.
What are polar coordinates in double integrals?
Sometimes it's easier to switch from Cartesian (x, y) to Polar (r, θ) coordinates, especially for circular regions. The differential area element dA becomes r dr dθ.
Is this calculator free?
Yes, our tools are completely free to use.