Triple Integral Calculator

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.

∫⁶₅∫⁴₃∫²₁ (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,

∫⁶₅∫⁴₃∫²₁ (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.

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