Double Integral Calculator

Enter the function f(x, y) to calculate the double integral (antiderivative) with this calculator.

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Double Integral Calculator

This calculator evaluates definite or indefinite double integrals of two-variable functions f(x, y). It provides step-by-step solutions and allows you to change the order of integration (dxdy or dydx) for easier computation.

What Is a Double Integral?

A double integral is used to calculate the volume under a surface, areas, mass, or flux over a 2D region. It integrates a function of two variables, f(x, y), over a specified region R.

Notation:

∬_R f(x, y) dA

Expanded Form:

∬_R f(x, y) dx dy or ∬_R f(x, y) dy dx

Where:

∬ = double integral
R = region of integration in the xy-plane
f(x, y) = function being integrated
dA = area element (dx dy or dy dx)

Steps to Solve Double Integrals

Define the Region R: Identify the limits for x and y.
Set Up the Integral: Write the double integral as ∬_R f(x, y) dA.
Evaluate the Inner Integral: Integrate with respect to one variable while treating the other as constant.
Evaluate the Outer Integral: Integrate the result with respect to the second variable.
Obtain the Final Result: The value of the double integral over R.

Example

Evaluate ∬ f(x, y) = x² + 3xy² + xy with limits 0 to 1 for both x and y.

Solution:

Step 1: Integrate with respect to x (inner integral)

∫₀¹ (x² + 3xy² + xy) dx = [x³/3 + (3/2)x²y² + (1/2)xy²]₀¹ = 1/3 + (3/2)y² + (1/2)y

Step 2: Integrate with respect to y (outer integral)

∫₀¹ (1/3 + 3/2 y² + 1/2 y) dy = [1/3 y + 1/2 y³ + 1/4 y²]₀¹ = 1/3 + 1/2 + 1/4 = 13/12

✅ Final Result: 13/12

For triple integrals, see our Triple Integral Calculator.

How To Use the Double Integral Calculator

Enter the function f(x, y)
Fill in upper and lower bounds for x and y
Select the order of integration (dxdy or dydx)
Click "Calculate"
Optional: Use the camera icon to upload an equation image or the keyboard icon for virtual input
Optional: Press "Load Example" for a sample calculation

FAQs

What are double integrals used for?

Volume under a surface: Compute the volume bounded by a surface over a region.
Average value: Compute the average of f(x, y) over a region.
Mass distribution: Calculate mass when density is ρ(x, y).
Flux calculation: Compute flux of a vector field over a region.