Triple integral calculator
Triple integral calculator is used to integrate the three-variable functions. The three-dimensional integration can be calculated by using our triple integral solver. It takes three different variables of integration to integrate the function.
How does the triple integration calculator work?
Follow the below steps to calculate the triple integral.
- First of all, select the definite or indefinite option.
- Enter the three-variable function into the input box.
- To enter the mathematical symbols, use the keypad icon
. - In the case of definite integral, enter the upper and lower limits of all the variables.
- Select the order of variables i.e., dxdydx, dydxdz, etc.
- Hit the calculate button to get the result.
- To enter a new function, press the reset button.
What is triple integral?
The triple integral is used to find the mass of a volume of a body that has variable density. It is similar to a double integral but in three dimensions. It integrates the given function over three-dimensional space.
Types of the triple integral are:
- Triple definite integral
- Triple indefinite integral
The equation of the triple definite integral is given below.
\(\int \int _B\int f\left(x,y,z\right)dV=\int _e^f\int _c^d\int _a^bf\left(x,y,z\right)dxdydz\)
The equation of triple indefinite integral is
\(\:\int \int \int f\left(x,y,z\right)dV=\int \int \int f\left(x,y,z\right)dxdydz\)
In the equations of the triple integral.
- f(x, y, z) is a three-variable function.
- a, b, c, d, e, and f are the upper and lower limits of x, y, and z.
- dx, dy, and dz are the integration variables of the given function.
How to evaluate triple integral problems?
Following are a few examples of triple integrals solved by our triple integrals calculator.
Example 1: For definite integral
Find triple integral of 4xyz, having limits x from 0 to 1, y from 0 to 2, and z from 1 to 2.
Solution
Step 1: Write the three-variable function along with the integral notation.
\( \int _1^2\int _0^2\int _0^14xyz\:dxdydz\:\:\:\)
Step 2: Integrate the three variable function w.r.t x.
\( \int _1^2\int _0^2\left(\int _0^14xyz\:dx\right)dydz\:\:\:\)
\( \int _1^2\int _0^2\left(4yz\int _0^1x\:dx\right)dydz\:\:\:\)
\( \int _1^2\int _0^2\left(4yz\left[\frac{x^{1+1}}{1+1}\right]^1_0\right)dydz\:\:\:\)
\( \int _1^2\int _0^2\left(4yz\left[\frac{x^2}{2}\right]^1_0\right)dydz\:\:\:\)
\( \int _1^2\int _0^2\left(2yz\left[x^2\right]^1_0\right)dydz\:\:\:\)
\( \int _1^2\int _0^2\left(2yz\left[1^2-0^2\right]\right)dydz\:\:\:\)
\(\int _1^2\int _0^2\left(2yz\right)dydz\:\:\:\)
Step 3: Now integrate the above expression w.r.t y.
\(\int _1^2\left(\int _0^22yz\:dy\right)dz\:\:\:\)
\(\int _1^2\left(2z\int _0^2y\:dy\right)dz\:\:\:\)
\(\int _1^2\left(2z\left[\frac{y^{1+1}}{1+1}\right]_0^2\right)dz\)
\(\int _1^2\left(2z\left[\frac{y^2}{2}\right]_0^2\right)dz\:\:\:\)
\(\int _1^2\left(z\left[y^2\right]_0^2\right)dz\:\:\:\)
\(\int _1^2\left(z\left[2^2-0^2\right]\right)dz\:\:\:\)
\(\int _1^2\left(z\left[4-0\right]\right)dz\:\:\:\)
\(\int _1^24z\:dz\:\:\:\)
Step 4: Integrate the above expression w.r.t z.
\(\int _1^24z\:dz\:\:\:\)
\(4\int _1^2z\:dz\:\:\:\)
\(4\left[\frac{z^{1+1}}{1+1}\right]_1^2\:\:\:\)
\(4\left[\frac{z^2}{2}\right]_1^2\:\:\:\)
\(2\left[z^2\right]_1^2\:\:\:\)
\(2\left[2^2-1^2\right]\:\:\:\)
\(2\left[4-1\right]\:\:\:\)
\(2\left[3\right]\:\:\:\)
\(6\)
Step 5: Now write the given function with the result.
\(\int _1^2\int _0^2\int _0^14xyz\:dxdydz=6\)
Example 2: For indefinite integral
Find triple integral of \(6x^2yz\) with respect to x, y, and z.
Solution
Step 1: Write the three-variable function along with the integral notation.
\( \int \int \int \:6x^2yz\:dxdydz\)
Step 2: Integrate the three variable function w.r.t x.
\( \int \int \left(\int \:\:6x^2yz\:dx\right)dydz\)
\( \int \int \left(6yz\int x^2\:dx\right)dydz\)
\( \int \:\int \:\left(6yz\left[\frac{x^{2+1}}{2+1}\right]+C\right)dydz\)
\( \int \:\int \:\left(6yz\left[\frac{x^3}{3}\right]+C\right)dydz\)
\(\int \:\int \:\left(2yz\left[x^3\right]+C\right)dydz\)
\(\int \:\int \left(2x^3yz+C\right)\:dydz\)
Step 3: Now integrate the above expression w.r.t y.
\(\int \:\:\left(\int \:2x^3yz\:dy+\int \:C\:dy\right)dz\)
\(\int \:\left(2x^3z\:\int \:y\:dy\:+C\int dy\right)dz\)
\( \int \:\left(2x^3z\:\left[\frac{y^{1+1}}{1+1}\right]+Cy+C\right)dz\)
\( \int \:\left(2x^3z\:\left[\frac{y^2}{2}\right]+Cy+C\right)dz\)
\( \int \:\left(x^3z\:\left[y^2\right]+Cy+C\right)dz\)
\( \int \left(x^3y^2z+Cy+C\right)dz\)
Step 4: Integrate the above expression w.r.t z.
\( \int \left(x^3y^2z+Cy+C\right)dz\)
\( \int \:x^3y^2z\:dz+\int \:Cy\:dz+\int \:Cdz\)
\( x^3y^2\int \:z\:dz+Cy\int \:dz+C\int \:dz\)
\( x^3y^2\left[\frac{z^{1+1}}{1+1}\right]+Cyz+Cz+C\)
\( x^3y^2\left[\frac{z^2}{2}\right]+Cyz+Cz+C\)
\(\frac{x^3y^2z^2}{2}+Cyz+Cz+C\)
References
- What is a triple integral? | Krista King Math | Online math help.
- Triple integral examples | Math Insight (n.d.).
