Triple Integrals In this section, we will learn about: Triple integrals.

Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.

Da Nang-02/2015

Lecturer: Ho Xuan Binh

In this section, we will learn about:

Triple integrals.

Triple Integrals  

Just as we defined single integrals for functions

of one variable and double integrals for

functions of two variables, so we can define

triple integrals for functions of three variables.

Triple Integrals s  

Let’s first deal with the simplest case where f is

defined on a rectangular box:

B  x y z a x b c y d r z s      

Triple Integrals s  

The first step is

to divide B into

sub-boxes—by

dividing:

The interval [a, b] into l

subintervals [x i-1 , x i]

of equal width Δx.

[c, d] into m subintervals of

width Δy.

[r, s] into n subintervals of

width Δz.

Triple Integrals  

The planes through the

endpoints of these

subintervals parallel to

the coordinate planes

divide the box B into

lmn sub-boxes

 1,  1,  1, 

ijk i i j j k k

B  x  x    y  y    z  z

Each sub-box has volume ΔV = Δx Δy Δz

Then, we form the triple Riemann

sum

1 1 1

, ,

l m n

ijk ijk ijk

i j k

  



  

where the sample point

is in Bijk.  xijk* , yijk* , zijk* 

Triple Integrals  

Triple Integrals in Cylindrical Coordinates  

The triple integral of f over the box B

is:

 * * * 

, ,

, ,

B

ijk ijk ijk

l m n

f x y z dV

 



  

if this limit exists

Again, the triple integral always exists if f

is continuous

We can choose the sample point to be any point in the sub-box

However, if we choose it to be the point

(x i , y j , z k) we get a simpler-looking expression:

1 1 1

l m n

B

 

  



Triple Integrals 

Thank you for your attention