Answer:

Now that we know how to integrate over a two-dimensional REGION we need to move on to integrating over a three-dimensional region. We used a double INTEGRAL to integrate over a two-dimensional region and so it shouldn’t be too surprising that we’ll use a triple integral to integrate over a three dimensional region. The notation for the GENERAL triple integrals is,

∭

E

f

(

X

,

y

,

z

)

d

V

∭Ef(x,y,z)dV

Let’s start simple by integrating over the box,

B

=

[

a

,

b

]

×

[

c

,

d

]

×

[

r

,

s

]

B=[a,b]×[c,d]×[r,s]

Note that when using this notation we list the

x

x

’s first, the

y

y

’s second and the

z

z

’s third.

The triple integral in this case is,

∭

B

f

(

x

,

y

,

z

)

d

V

=

∫

s

r

∫

d

c

∫

b

a

f

(

x

,

y

,

z

)

d

x

d

y

d

z

∭Bf(x,y,z)dV=∫rs∫cd∫abf(x,y,z)dxdydz

Note that we integrated with respect to

x

x

first, then

y

y

, and finally

z

z

here, but in fact there is no reason to the integrals in this order. There are 6 DIFFERENT possible orders to do the integral in and which order you do the integral in will depend upon the function and the order that you feel will be the easiest. We will get the same answer regardless of the order however.

Let’s do a quick example of this type of triple integrals.

Step-by-step explanation:

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