Double Integral Calculator with Steps

Function:

This double integral calculator integrates a function of two variables, f(x, y), over a specified two-dimensional region R. It evaluates both definite integrals (with specified numerical or variable bounds) and indefinite double integrals step-by-step. You can also switch the order of integration between (dx, dy) and (dy, dx) as needed.

What Is a Double Integral?

A double integral is used to integrate a function of two variables, such as f(x, y), and it is calculated over a two-dimensional region “R”. For a definite integral, the region is defined by the limits of the integration. It helps to find volume under a surface, areas, mass, or flux associated with a 2D region.

∬_R f(x, y) dA

In iterated form, you can write a double integral as:

∬_R f(x, y) dx dy

∬_R f(x, y) dy dx

Where:

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

Key Interpretations:

Volume Under a Surface: If f(x, y)0 over the region R, the volume is calculated using the double integral between the surface z = f(x, y) and the xy-plane
Area of a Region: When f(x, y)=1, in this case, the area of the region “R” is calculated by the double integral
Physical and Statistical Applications: Double integrals are also used to calculate mass, center of mass, moments of inertia, and joint probability distributions

How To Solve Double Integrals?

Follow the steps mentioned below to evaluate double integrals.

The example below shows how each step works.

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

Step 1: Define the Region R

The limits for both variables are from 0 to 1:

0 ≤ x ≤ 1
0 ≤ y ≤ 1

Step 2: Set Up the Integral

Write the double integral using the given limits:

∫₀¹ ∫₀¹ (x² + 3xy² + xy) dx dy

Step 3: Evaluate the Inner Integral

Integrate with respect to x while keeping y as a constant:

∫₀¹ (x² + 3xy² + xy) dx

= [x³/3 + (3/2)x²y² + (1/2)x²y]₀¹

= 1/3 + (3/2)y² + (1/2)y

Step 4: Evaluate the Outer Integral

Integrate the resulting expression with respect to y:

∫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

Step 5: Obtain the Final Result

= 13/12

How To Use the Double Integral Calculator?

Follow these simple steps to calculate the double integral of functions f(x,y):

Enter the Function: Type, copy/paste, or insert an image of a two-variable function f(x, y) into the input field
Set the Limits (Optional): Fill in the upper and lower limits for both x and y for a definite integral. Leave the limits empty for an indefinite integral
Set the Order of Integration: Use the preferred sequence of integration, either dx dy or dy dx
Click Calculate: Hit the "CALCULATE" button to process your input
View the Result: See the step-by-step evaluation of the double integral
Optional: Press "Load Example" for a sample calculation

For quick calculations over 2D regions, use our double integral solver. But if you need to calculate volume or mass across 3D space, try our Triple Integral Calculator.

FAQ's:

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 the flux of a vector field over a region

What functions can this calculator handle?

This double integration calculator can handle a variety of two-variable functions involving x and y, including polynomial, exponential, trigonometric, and logarithmic functions.

Can I reverse the order of integration?

Yes. Our double integrals calculator allows switching between dx dy and dy dx.

Does the order of integration matter?

No. By Fubini's Theorem, the order of integration (dx dy to dy dx) doesn’t matter if the function is continuous over the region R. You can compute either order (dx first or dy first) and get the same result.

∬_R f(x, y) dx dy = ∬_R f(x, y) dy dx

Note:

If the region is not a rectangle, then you may have to rewrite the limits after switching the order of integration.

What Is Fubini's Theorem?

Fubini’s theorem states that if a function f(x,y) is continuous over a region R. Then changing the order of integration with respect to “x” or “y” doesn’t affect the final result.

∬_R f(x, y) dx dy = ∫ (∫ f(x, y) dy) dx = ∫ (∫ f(x, y) dx) dy

When should I use polar coordinates in double integrals?

There are certain conditions for using polar coordinates:

The region of integration is circular, a disk, or bounded by arcs (e.g., x² + y² = a²).
The integral contains x² + y² terms, simplifying via r² = x² + y²
The area element can be converted from dA = dx dy to dA = r dr dθ.

What if I don’t know the bounds?

Definite integrals require numerical or variable bounds. If you are calculating an indefinite double integral, simply leave the bounds fields of our double integral calculator empty to compute the antiderivative symbolically.

Double Integral Calculator