# Triple Integral Calculator

To use a triple integral calculator, select the type of integral, enter the function and limit values of each variable, and click calculate button

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## Triple Integral Calculator

Triple Integral Calculator is used to find the integration of triple variable functions. This calculator is used to calculate the volume, and mass of three-dimensional objects.

## Define Triple Integral

In Calculus, the working of the triple integral is similar to single integral & double integral but it is used for three-dimensional space. It is used to evaluate the volume and also determines mass.

The triple integral function is expressed as:

∫∫∫ f (x,y, z) dxdydz

## How to calculate triple integral?

Here are some examples to understand the topic easily.

### Example 1: For Definite Integral

Calculate the triple integral of function f(x) = (x + y +z) having boundary values (2, 1), (4, 3), and (6, 5) of x, y, & z respectively with respect to dxdydz.

Solution:

Step 1: Write the given expression along with limit values.

654321 (x + y + z) dx dy dz

Step 2: Take the integral of the given function w.r.t "x".

= ∫21 (x+ y+ z) dx

= |x2/2 + x(y+z) |21

= (2y + 2z + 2) – (y + z + 1/2)

= y + z + 3/2

Step 3: Now take the integral of the given function w.r.t "y".

= ∫43 (y + z + 3/2) dy

= |y2/2 + y(z+3/2) |43

= (4z + 14) – (3z + 9)

= z + 5

Step 4: Now integrate the above expression w.r.t "z".

= ∫65 (z+5) dz

= |z2/2 + 5z|65

= (36/2 + 5(6)) – (52/2 + 5(5))

= 48 – 37.5

= 10.5

Hence,

654321 (x + y + z) dx dy dz = 10.5

### Example 2: For Indefinite Integral

Calculate the integral of function f(x) = (3x + 4y + 5z) w.r.t: dxdydz

Solution

Step 1: Write the given expression.

∫∫∫ (3x+ 4y+ 5z) dx dy dz ... (i)

Step 2: Take the integral of the given function w.r.t "x".

= ∫ (3x+ 4y+ 5z) dx

= 3x2/2 + x(4y + 5z)

Put in (i)

= ∫∫ 3x2/2 + x(4y+5z) dydz ... (ii)

Step 3: Now take the integral of the given function w.r.t "y".

= ∫∫ 3x2/2 + x(4y+5z) dy

= 2xy2 + y (3x2/2 + 5xz)

Step 4: Now integrate the above expression w.r.t "z".

= ∫ 2xy2 + y (3x2/2 + 5xz) dz

= 5xyz2/2 + z (3x2y/2+2xy2) + C

Here are some other results of the triple integral.

 Function Triple integral 4x+6y+7z 7xyz2/2 + z (2x2y + 3xy2) + C 7x+5y2+z xyz2/ 2 + z (7x2y/2 + 5xy3/3) + C (x+y)/z (x2y/2 + xy2/2) log(z) + C 4y+z+x xyz2/2 + z (x2y/2 + 2xy2) + C 3x(y+z) 3x2y2z/4 + 3x2yz2/4 + C 6z2-7y+2x 2xyz3 + (x2y – 7xy2/2) + C 5y+3z/x 5xy2z/2 + 3yz2log(x) /2 + C

You can cross-check these results by using our triple integral calculator.