Double Integral Calculator

Calculate double integrals step-by-step with this powerful calculator. Perfect for students and professionals in calculus, physics, and engineering.

Function f(x,y):

Lower limit for x:

Upper limit for x:

Lower limit for y:

Upper limit for y:

Integration Method:

Symbolic Numerical

Double Integral Result: 0

\iint\limits_{0}^{1}\iint\limits_{0}^{1} (x^2 + y^2) \,dx\,dy

Calculation Steps:

1. Integrate with respect to x:

\int (x^2 + y^2) \,dx = \frac{x^3}{3} + xy^2 + C

2. Evaluate from x=0 to x=1:

\left[ \frac{x^3}{3} + xy^2 \right]_{0}^{1} = \frac{1}{3} + y^2

3. Integrate with respect to y:

\int \left( \frac{1}{3} + y^2 \right) \,dy = \frac{y}{3} + \frac{y^3}{3} + C

4. Evaluate from y=0 to y=1:

\left[ \frac{y}{3} + \frac{y^3}{3} \right]_{0}^{1} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}

About Double Integral Calculator

A double integral calculator is a powerful tool that computes the integral of a function of two variables over a two-dimensional region. Double integrals are fundamental in multivariable calculus and have applications in physics, engineering, probability, and more.

What is a Double Integral?

A double integral extends the concept of a single integral to functions of two variables. It represents the volume under the surface defined by the function f(x,y) over a region R in the xy-plane.

\iint\limits_{R} f(x,y) \,dA

Iterated Integrals

Double integrals are typically evaluated as iterated integrals, where we integrate first with respect to one variable while treating the other as constant, and then integrate the result with respect to the second variable.

\int_{a}^{b} \left( \int_{c}^{d} f(x,y) \,dy \right) dx \quad \text{or} \quad \int_{c}^{d} \left( \int_{a}^{b} f(x,y) \,dx \right) dy

Applications of Double Integrals

Double integrals have numerous applications across mathematics and science:

Volume Calculation: Compute the volume under a surface and above a region in the xy-plane
Area Calculation: Find the area of a region by integrating 1 over that region
Physics: Calculate center of mass, moments of inertia, and gravitational potential
Probability: Determine joint probability distributions in statistics
Engineering: Solve problems in fluid mechanics, heat transfer, and electromagnetism

Types of Double Integrals

Our calculator supports different types of double integrals:

Rectangular Regions

When the region of integration is a rectangle [a,b]×[c,d], the double integral can be written as:

\int_{a}^{b} \int_{c}^{d} f(x,y) \,dy\,dx

General Regions

For more complex regions, the limits of integration may be functions rather than constants:

\int_{a}^{b} \int_{g_1(x)}^{g_2(x)} f(x,y) \,dy\,dx

Calculation Methods

Our calculator provides two approaches to compute double integrals:

Symbolic Integration

Finds exact solutions using antiderivatives and the Fundamental Theorem of Calculus. This method provides precise results when possible.

Numerical Integration

Uses approximation techniques like the Trapezoidal rule or Simpson's rule when symbolic integration is difficult or impossible. This is useful for complex functions or when only numerical results are needed.

Limitations

While powerful, double integral calculators have some limitations:

Symbolic integration may fail for highly complex functions
Numerical integration provides approximations, not exact values
Some regions may require changing the order of integration
Improper integrals may require special handling