Double Integral Calculator (General Region)

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

This Double Integral Calculator (General Region) integrates a function over a region whose upper and lower edges are curves rather than straight lines, and shows the working line by line. The inner limits depend on the outer variable, so the inner integral must be done first and its result still carries that variable. Only then do the outer limits apply. Following the same pattern every time keeps the order of integration fixed.

Step-by-step method

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

Formula:

\(\iint_R f\,dA = \int_{a}^{b}\!\int_{g_{1}(x)}^{g_{2}(x)} f(x,y)\,dy\,dx\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

In this problem:

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

Step 2

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

In this problem:

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

Step 3

Step 3 - Evaluate the inner integral at the y-curves.

In this problem:

\(\int_{0}^{x} x y\,dy = \frac{x^{3}}{2}\)

Step 4

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

In this problem:

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

Step 5

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

In this problem:

\(\iint_R f\,dA = \frac{1}{8} \approx 0.12\)

Final Answer

\(\iint_R f\,dA = \frac{1}{8} \approx 0.12\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

In this problem:

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

Step 2

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

In this problem:

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

Step 3

Step 3 - Evaluate the inner integral at the y-curves.

In this problem:

\(\int_{x^{2}}^{x} x + y\,dy = \frac{x^{2} \left(- x^{2} - 2 x + 3\right)}{2}\)

Step 4

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

In this problem:

\(\int_{0}^{1} \frac{x^{2} \left(- x^{2} - 2 x + 3\right)}{2}\,dx\)

Step 5

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

In this problem:

\(\iint_R f\,dA = \frac{3}{20}\)

Final Answer

\(\iint_R f\,dA = \frac{3}{20}\)

Example Problems

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