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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
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.
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