Double integral calculator with steps
Double integral calculator is used to integrate the double variable functions. This second integral calculator integrates the 2-D function with respect to corresponding integrating variables with steps. It solves the double integral by using two methods.
- Definite integral
- Indefinite integral
How does double antiderivative calculator work?
Follow the below steps to integrate double variable functions.
- First of all, select the definite or indefinite option.
- Input the double variable function f(x,y).
- Use the keypad icon to input the math symbols i.e., +, -, ^, etc.
- In the case of the definite integral, write the upper and the lower limits of both the integrating variables.
- Chose the integrating variables.
- Click the calculate button to get the result of the double variable function.
- Press the clear button to recalculate a new double variable function.
Note: Use inf for infinity, -inf for negative infinity, and pi for the π.
What is the double integral?
A way to integrate over a two-dimensional area is known as double integrals. It also computes the volume under a surface. In simple words, double integrals are used to integrate a double variable function with respect to its variables.
Double integrals used double integral notations to integrate the given function. It integrates the double variable function for double definite integrals or double indefinite integrals. Following are the equations of double integrals.
The general equation of the double definite integrals is:
\(\int _{y_1}^{y_2}\int _{x_1}^{x_2}f\left(x,y\right)dxdy\)
The general equation of double indefinite integral is:
\(\int\int f\left(x,y\right)dxdy\)
- In these equations, f(x,y) is a double variable function or integrand.
- \(x_1\&y_1\) are the lower limits and \(x_2\&y_2\) are the upper limits of double variable functions.
- The dx & dy are the integrating variables of the double integral function.
How to calculate the double integrals?
Here are some examples of multivariable functions solved by our double integration calculator.
Example 1: For the definite integral
Integrate the double variable function \( xy^2\) with respect to x and y having X limits from 2 to 3 and Y limits from 1 to 4.
Solution
Step 1: Use the integral notation to write the given double variable function.
\(\int _1^4\int _2^3\left(xy^2\right)\:dxdy\)
Step 2: Now integrate the double variable function with respect to x.
\(\int _1^4\left(\int _2^3\left(xy^2\right)dx\right)dy\)
\(\int _1^4\left(y^2\int _2^3\left(x\right)dx\right)dy\)
\(\int _1^4\left(y^2\left(\frac{x^{1+1}}{1+1}\right)^3_2\right)dy\)
\(\int _1^4\left(\frac{y^2}{2}\left(x^2\right)^3_2\right)dy\)
Apply the upper and the lower limits by using the fundamental theorem of calculus.
\(\int _1^4\left(\frac{y^2}{2}\left(3^2-2^2\right)\right)dy\)
\(\int _1^4\left(\frac{y^2}{2}\left(9-4\right)\right)dy\)
\(\int _1^4\left(\frac{y^2}{2}\left(5\right)\right)dy\)
\(\int _1^4\frac{5y^2}{2}dy\)
Step 3: Now integrate the double variable function with respect to y.
\(\frac{5}{2}\int _1^4y^2dy\)
\(\frac{5}{2}\left[\frac{y^{2+1}}{2+1}\right]^4_1\)
\(\frac{5}{2}\left[\frac{y^3}{3}\right]^4_1\)
\(\frac{5}{6}\left[y^3\right]^4_1\)
\(\frac{5}{6}\left[4^3-1^3\right]\)
\(\frac{5}{6}\left[64-1\right]\)
\(\frac{5}{6}\left[63\right]\)
\(\frac{5}{2}\left[21\right]\)
\(\frac{105}{2}\)
Step 4: Write the given double variable function with the result.
\(\int _1^4\int _2^3\left(xy^2\right)\:dxdy=\frac{105}{2}\)
Example 2: For indefinite integral.
Integrate the double variable function \( x^2y\) with respect to y and x.
Solution
Step 1: Use the integral notation to write the given double variable function.
\(\int \:\int \:x^2y\:dydx\)
Step 2: Now integrate the double variable function with respect to x.
\(\int \left(\int \:x^2y\:dx\right)dy\)
\(\int \left(y\int \:\text{x}^2\:dx\right)dy\)
\(\int \:\left(y\left(\frac{x^{2+1}}{2+1}\right)+C_1\right)dy\)
\(\int \:\left(y\left(\frac{x^3}{3}\right)+C_1\right)dy\)
\(\int \left(\frac{x^3y}{3}+C_1\right)dy\)
Step 3: Now integrate the double variable function with respect to y.
\(\int \left(\frac{x^3y}{3}\right)dy+\int C_1dy\)
\(\frac{x^3}{3}\int ydy+\int C_1dy\)
\(\frac{x^3}{3}\left(\frac{y^2}{2}\right)+C_1y+C_2\)
\(\frac{x^3y^2}{6}+C_1y+C_2\)
Step 4: Write the given double variable function with the result.
\(\int \int x^2y\:dydx=\frac{x^3y^2}{6}+C_1y+C_2\)