|
|
In a Nut Shell:
No
matter the type of region of integration each triple integral contains
three distinct parts. They include the inner
integral, the middle integral, and the outer
integral. The table below illustrates these parts
for a Type 1 region.
|
|
The “inner integral” of ∫ ∫ [ ∫
f(x,y,z) dz ] dA is the one in
the brackets.
When evaluating this
part of the triple integral the independent variables, x
and y (within
dA), are held constant as if the function f(x,y,z) only depended on z. The limits of integration
for the inner integral
are from z = u1(x,y) to z = u2(x,y).
z = u2 (x,y)
Inner integral ∫ f(x,y,z) dz
z = u1(x,y)
|
|
The “middle integral” of
∫ [ ∫ ∫ f(x,y,z) dz dx ] dy is the one in the brackets.
When evaluating this
portion of the triple integral the independent variable y is held
constant as if the
function f(x,y,z)
only depended on x.
x = v2(y) z= u2 (x,y)
Middle integral ∫ [ ∫ f(x,y,z) dz ] dx dy
x
= v1(y) z = u1(x,y)
|
|
Finally the “outer integral” is ∫ ∫
∫ f(x,y,z) dz dx dy
. When evaluating this final portion
of
the integral the last
variable, in this case, y, must go from one constant to another. i.e.
y
= b x = v2(y) z = u2 (x,y)
Outer integral
∫ ∫
[ ∫ f(x,y,z) dz ] dx dy
y = a
x = v1(y) z
= u1(x,y)
|
|