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" button to find the value of the double integral.
- Step 4: Click on the "Reset" button to clear the fields and enter new values.
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 Integral Calculator
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:
- \(\int_{3.2}^{5.5}\int_{6}^{7} \frac{x^{2}y}{3} dydx\)
- \(\int_{2}^{5}\int_{8}^{13} [x^{2}y + xy^{2}]dxdy\)