Calculators / Calculus 3 / Double Integral Calculator (Rectangular Region)

Double Integral Calculator (Rectangular Region)

Published on: August 23, 2026

This Double Integral Calculator (Rectangular Region) helps you evaluate double integrals over rectangular regions in the plane. The calculator uses the constant bounds for x and y, sets up the iterated integral correctly, and then works through the integration step by step. This makes it easier to understand the region, the limits, and the order of integration. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Step-by-step method

  1. Identify the integrand and read the inner and outer bounds for the rectangular region.
  2. Evaluate the inner integral first with respect to the inner variable.
  3. Apply the inner bounds and simplify to get a function of the outer variable.
  4. Integrate that result with respect to the outer variable.
  5. Apply the outer bounds and simplify the final answer.

Formulas:

Rectangular-region double integral

∫ba∫dcf( x, y ) dy dx

Inner integral

∫dcf( x, y ) dy = [F( x, y )]dc

Outer integral

∫bag( x ) dx = [G( x )]ba

Example 1: f(x,y)=x+y; y:[0,2]; x:[1,3]

Step 1A - Identify the integrand.

In this problem: Read the function being integrated.

f( x, y ) = x + y

Step 1B - Identify the inner bounds.

In this problem: Read the inner variable and its interval in the rectangular region.

Inner: y from 0 to 2

Step 1C - Identify the outer bounds.

In this problem: Read the outer variable and its interval in the rectangular region.

Outer: x from 1 to 3

Step 2A - Set up the inner integral.

In this problem: Integrate first with respect to the inner variable.

I₁ = ∫20x + y dy

Step 2B - Find an antiderivative for the inner integral.

In this problem: Integrate with respect to y.

∫ x + y dy =
y(y + 2x)
2

Step 2C - Apply the inner bounds.

In this problem: Use the antiderivative and evaluate at the limits.

I₁ = [
y(y + 2x)
2
]20

Step 2D - Simplify the inner result.

In this problem: This becomes a function of x.

I₁ = 2(1 + x)

Step 3A - Set up the outer integral.

In this problem: Now integrate the inner result with respect to the outer variable.

I = ∫312(1 + x) dx

Step 3B - Find an antiderivative for the outer integral.

In this problem: Integrate with respect to x.

∫ 2(1 + x) dx = x(2 + x)

Step 3C - Apply the outer bounds.

In this problem: Evaluate the outer antiderivative at the limits.

I = [x(2 + x)]31

Step 3D - Write the final answer.

In this problem: This is the value of the rectangular-region double integral.

I = 12

Final answer: 12

Example 2: f(x,y)=x*y; y:[1,3]; x:[0,2]

Step 1A - Identify the integrand.

In this problem: Read the function being integrated.

f( x, y ) = xy

Step 1B - Identify the inner bounds.

In this problem: Read the inner variable and its interval in the rectangular region.

Inner: y from 1 to 3

Step 1C - Identify the outer bounds.

In this problem: Read the outer variable and its interval in the rectangular region.

Outer: x from 0 to 2

Step 2A - Set up the inner integral.

In this problem: Integrate first with respect to the inner variable.

I₁ = ∫31xy dy

Step 2B - Find an antiderivative for the inner integral.

In this problem: Integrate with respect to y.

∫ xy dy =
xy2
2

Step 2C - Apply the inner bounds.

In this problem: Use the antiderivative and evaluate at the limits.

I₁ = [
xy2
2
]31

Step 2D - Simplify the inner result.

In this problem: This becomes a function of x.

I₁ = 4x

Step 3A - Set up the outer integral.

In this problem: Now integrate the inner result with respect to the outer variable.

I = ∫204x dx

Step 3B - Find an antiderivative for the outer integral.

In this problem: Integrate with respect to x.

∫ 4x dx = 2x2

Step 3C - Apply the outer bounds.

In this problem: Evaluate the outer antiderivative at the limits.

I = [2x2]20

Step 3D - Write the final answer.

In this problem: This is the value of the rectangular-region double integral.

I = 8

Final answer: 8