Triple integral calculator with steps
Triple integral calculator integrates the given multivariable functions with respect to its corresponding integrating variables. This triple integration calculator solves the given function in three dimensions.
How does this triple integral calculator work?
To integrate the three-variable function, follow the below steps.
- Select the type of integral i.e., definite or indefinite.
- Input the three-variable function.
- Use the keypad icon to enter math keys.
- Enter the upper and lower limits of all the integrating variables in case of the definite integral.
- Choose the integrating variable’s order.
- Press the load example button to use the sample examples.
- Press the calculate button.
- Hit the clear button to calculate another three-variable function.
What is triple integral?
Triple integrals are the correspondent of double integrals for 3-D. It is a way of adding up infinitely many infinitesimal quantities associated with points in a 3-D region. Triple integral is widely used to find the mass of the body that has a variable density.
It can calculate the triple variable function by using two methods.
(i) Definite integral
(ii) Indefinite integral
The general equation for integrating three-variable function by using definite integrals is:
\(\int \int \int f\left(x,y,z\right)dV=\int _{z_1}^{z_2}\int _{y_1}^{y_2}\:\int _{x_1}^{x_2}f\left(x,y,z\right)dxdydz\)
The general equation for integrating three-variable function by using indefinite integrals is:
\(\int \int \int f\left(x,y,z\right)dxdydz\)
- \( f\left(x,y,z\right)\) is the given three-variable function.
- \(x_1\&x_2\) are the upper and lower limits of x, \(y_1\&y_2\) are the upper and lower limits of y, and \(z_1\&z_2\) are the upper and lower limits of z.
- dx, dy, & dz are the integrating variables of the three-variables function.
How to calculate triple integrals?
Let’s take some examples to learn how to integrate the three-variable function.
Example
Integrate 3x+y+z with respect to x, y, & z having limits 0 to 1, 1 to 2, 2 to 3 respectively.
Solution
Step 1: Write the definite integral notation with the given function.
\(\int _2^3\int _1^2\int _0^1\left(3x+y+z\:\right)dxdydz\)
Step 2: Now integrate the definite integral with respect to x.
\(\int _2^3\int _1^2\left(\int _0^1\left(3x+y+z\:\right)dx\right)dydz\)
\(\int _2^3\int _1^2\left(\int _0^13x\:dx+\int _0^1y\:dx+\int _0^1z\:dx\right)dydz\)
\(\int _2^3\int _1^2\left(3\left[\frac{x^{1+1}}{1+1}\right]^1_0+y\left[x\right]^1_0+z\:\left[x\right]^1_0\right)dydz\)
\(\int _2^3\int _1^2\left(\frac{3}{2}\left[x^2\right]^1_0+y\left[x\right]^1_0+z\:\left[x\right]^1_0\right)dydz\)
\(\int _2^3\int _1^2\left(\frac{3}{2}\left[1^2-0^2\right]+y\left[1-0\right]+z\:\left[1-0\right]\right)dydz\)
\(\int _2^3\int _1^2\left(\frac{3}{2}+y+z\:\right)dydz\)
Step 3: Now integrate the definite integral with respect to y.
\( \int _2^3\left(\int _1^2\left(\frac{3}{2}+y+z\:\right)dy\right)dz\)
\( \int _2^3\left(\int _1^2\frac{3}{2}dy+\int _1^2y\:dy+\int _1^2z\:dy\right)dz\)
\( \int _2^3\left(\frac{3}{2}\left[y\right]^2_1+\left[\frac{y^{1+1}}{1+1}\right]^2_1+z\left[y\right]^2_1\right)dz\)
\( \int _2^3\left(\frac{3}{2}\left[2-1\right]+\left[\frac{y^2}{2}\right]^2_1+z\left[2-1\right]\right)dz\)
\(\int _2^3\left(\frac{3}{2}+\left[\frac{2^2-1^2}{2}\right]+z\right)dz\)
\(\int _2^3\left(\frac{3}{2}+\frac{3}{2}+z\right)dz\)
\(\int _2^3\left(3+z\right)dz\)
Step 4: Now integrate the definite integral with respect to z.
\(\int _2^3\left(3\right)dz\)+\(\int _2^3\left(z\right)dz\)
\(3\left[z\right]_2^3+\left[\frac{z^{1+1}}{1+1}\right]_2^3\)
\(3\left[3-2\right]+\left[\frac{z^2}{2}\right]_2^3\)
\(3\left[1\right]+\left[\frac{3^2-2^2}{2}\right]\)
\(3+\left[\frac{9-4}{2}\right]\)
\(3+\frac{5}{2}\)
\(5.5\)
Step 5: Now write the given function with the result.
\(\int _2^3\int _1^2\int _0^1\left(3x+y+z\:\right)dxdydz=5.5\)