# Double Integral Calculator

Double Integral Calculator calculates the value of a double integral. The area of a 2-dimensional figure can be determined with the help of double integrals. Double integration is represented by** '∫∫ '.**

## What is Double Integral Calculator?

Double Integral Calculator is an online tool that helps to integrate a given function and obtain the value of the double integral. Double integrals can be used to find the volume under a surface and the average value of a function with two variables. To use the ** Double Integral Calculator**, enter the values in the input boxes.

### Double Integral Calculator

## How to Use Double Integral Calculator?

Please follow the steps given below to find the value of the double integral using the online double integral calculator:

**Step 1:**Go to Cuemath's online Double Integral Calculator.**Step 2:**Enter the function as well as the limits in the given input boxes. From the drop-down list choose which variable will be integrated first.**Step 3:**Click on the**"Calculate"****Step 4:**Click on the**"Reset"**

## How Does Double Integral Calculator Work?

Integral calculus consists of certain different types of integrations such as simple integration, double integration, and triple integration. When we deal with a function in one variable, the integration is applied over an interval (one-dimensional space). Thus, when we have a function that depends on two variables, we essentially integrate it over a region (2-dimensional space).

If we have a double integral represented as \(\int_{c}^{d}\int_{a}^{b}f(x,y)dxdy\), then we use the following steps to find its value.

- We first solve the inner integral. As dx comes before dy thus, we will first integrate the function with respect to x. All the terms containing y will be treated as constants.
- The inner limits of the definite integral are applied. Now our function will be only in terms of y.
- Next, we solve the outer integral. This implies that we are integrating the function with respect to y.
- Apply the limits of the outer integral to get the final value.

As a note the integral value of \(\int_{c}^{d}\int_{a}^{b}f(x,y)dxdy\) will be equal to \(\int_{a}^{b}\int_{c}^{d}f(x,y)dydx\).

## Solved Examples on Double Integrals

**Example 1:** Find the double integral value of \(\int_{0}^{1}\int_{2}^{3}x^{3}ydxdy\) and verify it using the double integral calculator.

**Solution:**

I = \(\int_{0}^{1}\int_{2}^{3}x^{3}ydxdy\)

We first integrate the function with respect to x

I = \(\int_{0}^{1}[\int_{2}^{3}x^{3}ydx]dy\)

I = \(\int_{0}^{1}[\frac{x^{4}y}{4}]_{2}^{3}dy\)

Now we integrate the function with respect y

I = \(\int_{0}^{1}\frac{65y}{4}dy\)

I = \([\frac{65y^{2}}{8}]_{0}^{1}\textrm{}\)

I = 65/8

I = 8.125

**Example 2:** Find the double integral value of \(\int_{6}^{8.2}\int_{1}^{2} (xy - y)dydx\) and verify it using the double integral calculator.

**Solution:**

I = \(\int_{6}^{8.2}\int_{1}^{2} (xy - y)dydx\)

We first integrate the function with respect to y

I = \(\int_{6}^{8.2} [\frac{y^{2}x}{2} - \frac{y^{2}}{2}]_{1}^{2}dx\)

Now we integrate the function with respect x

I = \(\int_{6}^{8.2} [\frac{3x}{2} - \frac{3}{2}]dx\)

I = \([\frac{3x^{2}}{4} - \frac{3x}{2}]_{6}^{8.2}\)

I = 20.13

Similarly, you can use the double integral calculator to find the value of double integrals for the following:

- \(\int_{3.2}^{5.5}\int_{6}^{7} \frac{x^{2}y}{3} dydx\)
- \(\int_{2}^{5}\int_{8}^{13} [x^{2}y + xy^{2}]dxdy\)

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