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## 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

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