Solved Examples on Triple Integrals

Example 1. Evaluate the triple integral problem:

[Tex]\iiint k dV =\int_0^{z=12}\int_0^{y=12}\int_0^{x=12} k dx dy dz [/Tex]

Solution:

[Tex]\iiint k dV =\int_0^{z=12}\int_0^{y=12}\int_0^{x=12} k dx dy dz [/Tex]

⇒ [Tex]\iiint k dV =\int_0^{z=12}\int_0^{y=12} [/Tex]⇒ k [x] ₀ ^x=12 dy dz

⇒ [Tex]\iiint k dV = \int_0^{z=12}\int_0^{y=12} [/Tex] k [x]₀ ^x=12 dy dz

⇒ [Tex]\iiint k dV = \int_0^{z=12} [/Tex] 12k[y]₀ ^y=12 dz

⇒ [Tex]\iiint k dV [/Tex] = 144k [z]₀ ^z=12

⇒ [Tex]\iiint [/Tex] k dV = 1728 k

Example 2. Evaluate the triple integral problem

[Tex]\int_0^{z=8}\int_0^{y=6}\int_0^{x=4} k dx dy dz [/Tex]

Solution:

[Tex]\int_0^{z=8}\int_0^{y=6}\int_0^{x=4} k dx dy dz [/Tex]

= [Tex]\int_0^{z=8}\int_0^{y=6} [/Tex] k [x]₀ ^x=4 dy dz

= [Tex]\int_0^{z=8} [/Tex]4k[y]₀ ^y=6 dz

= 24k [z]₀ ^z=8

= 192 k

Example 3. Evaluate the triple integral problem

[Tex]\int_0^{z=3}\int_0^{y=2}\int_0^{x=4} 4x dx dy dz [/Tex]

Solution:

[Tex]\int_0^{z=3}\int_0^{y=2}\int_0^{x=4} 4x dx dy dz [/Tex]

= [Tex]\int_0^{z=3}\int_0^{y=2} [/Tex]4k [x²/2]₀ ^x=4 dy dz

= [Tex]\int_0^{z=3} [/Tex] 4k[8][y]₀ ^y=2 dz

= 64k [z]₀ ^z=3

= 192 k

Example 4. Evaluate the triple integral problem

[Tex]\int_0^{z=10}\int_0^{y=12}\int_0^{x=5} k dx dy dz [/Tex]

Solution:

[Tex]\int_0^{z=10}\int_0^{y=12}\int_0^{x=5} k dx dy dz [/Tex]

= [Tex]\int_0^{z=10}\int_0^{y=12} [/Tex] [x]₀ ⁵ dy dz

= [Tex]\int_0^{z=10} [/Tex] [y]₀ ¹² 5k dz

= [z]₀ ¹⁰ 60k

= 600k

Example 5. Evaluate the triple integral problem

[Tex]\int_0^{z=18}\int_0^{y=9}\int_0^{x=3} k dx dy dz [/Tex]

Solution:

[Tex]\int_0^{z=18}\int_0^{y=9}\int_0^{x=3} k dx dy dz [/Tex]

= [Tex]\int_0^{z=18}\int_0^{y=9} [/Tex] [x]₀ ^x=3 dy dz

= [Tex]\int_0^{z=18} [/Tex] [y]₀ ^y=9 3k dz

= [z] ₀ ^z=18 27k

= 486k

Triple Integrals: Integrals are an essential part of the mathematical world and hold great significance in today’s world. There are different types of integrals and each has its importance in mathematics. Some of these different types of integrals in mathematics are linear integrals, double integrals, triple integrals, etc.

Triple Integral is one of the types of multi integral of a function that involves three variables. Triple Integral in Calculus is the integration involving volume, hence it is also called Volume Integral and the process of calculating Triple Integral is called Triple Integration.

In this article, we will discuss triple integrals in detail along with their examples and representation and steps to solve multiple triple integral problems.

Read in Detail: Integrals