Double Integral Calculator (Rectangular Region)

Published on: July 21, 2026
Final Answer: Free • Full Steps: Pro

This Double Integral Calculator (Rectangular Region) integrates a function of two variables over a rectangle, and shows the working line by line. The inner integral is worked out first with the outer variable held constant, which turns it into an ordinary integral. Its result is then integrated again over the remaining limits. Following the same pattern every time keeps the two integrations from being run together.

Step-by-step method

  1. Set up the problem.
  2. Integrate with respect to x first (inner integral), treating y as constant.
  3. Evaluate the inner integral at the x-limits.
  4. Integrate the result with respect to y (outer integral).
  5. Evaluate at the y-limits to get the final value.

Formula:

\(\iint_R f\,dA = \int_{c}^{d}\!\int_{a}^{b} f(x,y)\,dx\,dy\)
See Example 1 Hide Example 1

Problem

\(\int_{0}^{2}\!\int_{0}^{1} x y\,dx\,dy\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Integrate with respect to x first (inner integral), treating y as constant.
  3. Evaluate the inner integral at the x-limits.
  4. Integrate the result with respect to y (outer integral).
  5. Evaluate at the y-limits to get the final value.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\int_{0}^{2}\!\int_{0}^{1} x y\,dx\,dy\)

Step 2

Step 2 - Integrate with respect to x first (inner integral), treating y as constant.

In this problem:

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

Step 3

Step 3 - Evaluate the inner integral at the x-limits.

In this problem:

\(\int_{0}^{1} x y\,dx = \frac{y}{2}\)

Step 4

Step 4 - Integrate the result with respect to y (outer integral).

In this problem:

\(\int \frac{y}{2}\,dy = \frac{y^{2}}{4} + C\)

Step 5

Step 5 - Evaluate at the y-limits to get the final value.

In this problem:

\(\iint_R f\,dA = 1\)

Final Answer

\(\iint_R f\,dA = 1\)
See Example 2 Hide Example 2

Problem

\(\int_{1}^{2}\!\int_{0}^{1} x^{2} + y\,dx\,dy\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Integrate with respect to x first (inner integral), treating y as constant.
  3. Evaluate the inner integral at the x-limits.
  4. Integrate the result with respect to y (outer integral).
  5. Evaluate at the y-limits to get the final value.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\int_{1}^{2}\!\int_{0}^{1} x^{2} + y\,dx\,dy\)

Step 2

Step 2 - Integrate with respect to x first (inner integral), treating y as constant.

In this problem:

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

Step 3

Step 3 - Evaluate the inner integral at the x-limits.

In this problem:

\(\int_{0}^{1} x^{2} + y\,dx = y + \frac{1}{3}\)

Step 4

Step 4 - Integrate the result with respect to y (outer integral).

In this problem:

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

Step 5

Step 5 - Evaluate at the y-limits to get the final value.

In this problem:

\(\iint_R f\,dA = \frac{11}{6} \approx 1.83\)

Final Answer

\(\iint_R f\,dA = \frac{11}{6} \approx 1.83\)

Example Problems

Click one to load it into the box and solve it.