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  • Recognize when a function of three variables is integrable over a rectangular box.
  • Evaluate a triple integral by expressing it as an iterated integral.
  • Recognize when a function of three variables is integrable over a closed and bounded region.
  • Simplify a calculation by changing the order of integration of a triple integral.
  • Calculate the average value of a function of three variables.

In Double Integrals over Rectangular Regions , we discussed the double integral of a function f ( x , y ) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f ( x , y , z ) of three variables over a rectangular solid box in space, 3 . Later in this section we extend the definition to more general regions in 3 .

Integrable functions of three variables

We can define a rectangular box B in 3 as B = { ( x , y , z ) | a x b , c y d , e z f } . We follow a similar procedure to what we did in Double Integrals over Rectangular Regions . We divide the interval [ a , b ] into l subintervals [ x i 1 , x i ] of equal length Δ x = x i x i 1 l , divide the interval [ c , d ] into m subintervals [ y i 1 , y i ] of equal length Δ y = y j y j 1 m , and divide the interval [ e , f ] into n subintervals [ z i 1 , z i ] of equal length Δ z = z k z k 1 n . Then the rectangular box B is subdivided into l m n subboxes B i j k = [ x i 1 , x i ] × [ y i 1 , y i ] × [ z i 1 , z i ] , as shown in [link] .

In x y z space, there is a box B with a subbox Bijk with sides of length Delta x, Delta y, and Delta z.
A rectangular box in 3 divided into subboxes by planes parallel to the coordinate planes.

For each i , j , and k , consider a sample point ( x i j k * , y i j k * , z i j k * ) in each sub-box B i j k . We see that its volume is Δ V = Δ x Δ y Δ z . Form the triple Riemann sum

i = 1 l j = 1 m k = 1 n f ( x i j k * , y i j k * , z i j k * ) Δ x Δ y Δ z .

We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.


The triple integral    of a function f ( x , y , z ) over a rectangular box B is defined as

lim l , m , n i = 1 l j = 1 m k = 1 n f ( x i j k * , y i j k * , z i j k * ) Δ x Δ y Δ z = B f ( x , y , z ) d V

if this limit exists.

When the triple integral exists on B , the function f ( x , y , z ) is said to be integrable on B . Also, the triple integral exists if f ( x , y , z ) is continuous on B . Therefore, we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, f is bounded on B and continuous except possibly on the boundary of B . The sample point ( x i j k * , y i j k * , z i j k * ) can be any point in the rectangular sub-box B i j k and all the properties of a double integral apply to a triple integral. Just as the double integral has many practical applications, the triple integral also has many applications, which we discuss in later sections.

Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as in the case of the double integral, we can have an iterated triple integral, and consequently, a version of Fubini’s thereom for triple integrals exists.

Fubini’s theorem for triple integrals

If f ( x , y , z ) is continuous on a rectangular box B = [ a , b ] × [ c , d ] × [ e , f ] , then

B f ( x , y , z ) d V = e f c d a b f ( x , y , z ) d x d y d z .

This integral is also equal to any of the other five possible orderings for the iterated triple integral.

Questions & Answers

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Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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