To calculate result you have to disable your ad blocker first.
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
Table of Contents:
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.
| 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.