Double integral calculator with steps
Double integral calculator is used to solve the antiderivative of two-variable functions. This double integration calculator finds 2-dimensional integration with steps.
It can calculate the problems of the double definite integral as well as a double indefinite integral.
How does the Double integration Calculator work?
Second integral calculator is an easy-to-use tool. Follow the below steps to calculate the second integral.
- Select the option first, definite or indefinite.
- Enter the two-variable function into the input box.
- Use the keypad icon
to enter mathematical symbols. - Enter the upper and lower limits of both the variables, in the case of the definite integral.
- Select the order of variables i.e., dxdy or dydx.
- Press the calculate key to get the result.
- Hit the reset key to enter a new function.
What is double integral?
A double integral is a type of integral having a two-variable function f(x, y) used to find the volume between the graph and a rectangular region of the xy-plane by taking an integral of an integral. The process of finding the double integral is known as double integration.
Types:
- Definite integrals
- Indefinite integrals
The equation of double definite integral:
\( \int _{y_1}^{y_2}\left(\int _{x_1}^{x_2}f\left(x,y\right)dx\right)dy\)
The equation can also be written in other way if you want to calculate the second variable first.
\(\int _{x_1}^{x_2}\left(\int _{y_1}^{y_2}f\left(x,y\right)dy\right)dx\)
But in the case of indefinite integral, the upper and lower limits are not used.
\(\int\int f\left(x,y\right)dxdy\)
In these equations,
- f(x, y) is the double variable function.
- \(x_1,\:x_2,\:y_1,\:and\:y_2\) are the upper and lower limits of double variable functions.
- And dx & dy are the variables of integration.
How to calculate the double integrals?
Following are a few examples of double integrals solved by our double integrals calculator.
Example 1: For the definite integral
Find the double integral of \(x^2+2y^2\) w.r.t x & y having limits from 1 to 3 for x and 2 to 4 for y.
Solution
Step 1: Write the given function along with the double integral notation.
\(\int _2^4\int _1^3\left(x^2+2y^2\right) dxdy\)
Step 2: Integrate the above expression w.r.t "x".
\(\int _2^4\left(\int _1^3\left(x^2+2y^2\right)dx\right)dy\)
\(\int _2^4\left(\int _1^3x^2dx+\int _1^3 2y^2dx\right)dy\)
\(\int _2^4\left(\left[\frac{x^{2+1}}{2+1}\right]^3_1+\left[2y^2x\right]^3_1\right)dy\)
\(\int _2^4\left(\frac{1}{3}\left[x^3\right]^3_1+2y^2\left[x\right]^3_1\right)dy\)
Apply the Fundamental theorem of calculus.
\(\int _2^4\left(\frac{1}{3}\left[3^3-1^3\right]+2y^2\left[3-1\right]\right)dy\)
\(\int _2^4\left(\frac{1}{3}\left[27-1\right]+2y^2\left[3-1\right]\right)dy\)
\(\int _2^4\left(\frac{26}{3}+4y^2\right)dy\)
Step 3: Now integrate w.r.t "y".
\(\int _2^4\frac{26}{3}dy+\int _2^44y^2dy\)
\(\left[\frac{26}{3}y\right]^4_2+\left[4\frac{y^{2+1}}{2+1}\right]^4_2\)
\(\frac{26}{3}\left[y\right]^4_2+\frac{4}{3}\left[y^3\right]^4_2\)
\(\frac{26}{3}\left[4-1\right]+\frac{4}{3}\left[4^3-2^3\right]\)
\(\frac{26}{3}\left[3\right]+\frac{4}{3}\left[64-8\right]\)
\(\left(26+\frac{4}{3}\left[56\right]\right)\)
\(26+\frac{224}{3}\)
\(\frac{302}{3}\)
\(100.67\)
Step 4: Now write the input with result.
\(\int _2^4\int _1^3\left(x^2+2y^2\right)dxdy=100.67\)
Example 2: For indefinite integral.
Find the double integral of xy with respect to x & y.
Solution
Step 1: Write the given function along with the double integral notation.
\( \int \int xy\ dxdy\)
Step 2: Integrate the above expression w.r.t "x".
\(\int \left(\int {xy}dx\right)dy \)
\( \int \left(y\int x dx\right)dy\)
\(\int \left(y\left(\frac{x^2}{2}\right)+c_1\right)dy\)
\( \int \left(\frac{x^2\text{y}}{2}+c_1\right)dy \)
Step 3: Now integrate w.r.t "y".
\( \frac{x^2}{2}\int \text{y dy}+\int c_1dy \)
\( \frac{x^2}{2}\left(\frac{y^2}{2}\right)+c_1y+c_2\)
\( \frac{x^2y^2}{4}+c_1y+c_2\)
Step 4: Now write the input with result.
\(\int \int xy dxdy=\frac{x^2y^2}{4}+c_1y+c_2\)
Resources
- Definition and formulas of double integral | Khan Academy (n.d.).
- Example of double integral | Study.com
