Summary of Multivariable Calculus MVC Lecture 5 for MATH2310 Engineering Calculus

MATH2310 – MVC Lecture 5


Triple integrals:
Let a , b , c , d , r , s ∈ R such that a ≤ b , c ≤ d , r ≤ s, then:
B=[ a , b ] × [ c , d ] ×[r , s ]
is a rectangular prism or box.
¿ ¿ ¿
If B is divided into lmn many equally sized sub-rectangles with ( x i , y j , z k ) in the
( i , j, k )th sub-rectangle. Then:
❑ n m l

∭ f ( x , y , z ) dV =l , mlim
,n → ∞
∑ ∑ ∑ f ( x i , y j , zk ) ∆ x ∆ y ∆ z
¿ ¿ ¿

B i=1 j=1 k=1


Fubuni’s theorem also applies to triple integrals.
❑ s d b

∭ f ( x , y , z ) dV =∫∫∫ f ( x , y , z ) dx dy dz
B r c a


The above integral may also be permuted.
Triple integrals may also be written as a double integral.
❑ ❑ ❑

∭ f ( x , y , z ) dV =∬∫ f ( x , y , z ) dz dA
B D C


where C is of the form C=[ z 0 ( x , y ) , z 1 ( x , y ) ].



Change of variables:
Let T be a transformation which takes R in uvw -space to xyz -space via the maps:
x=x ( u , v , w ) , y = y (u , v , w ) , z=z (u , v , w)
The Jacobian of T is defined to be:




| |
∂x ∂x ∂x
∂u ∂v ∂w
∂( x , y , z) ∂ y ∂y ∂y
=
∂(u , v , w) ∂ u ∂v ∂w
∂z ∂z ∂z
∂u ∂v ∂w

This transformation is given by:


| |
❑ ❑
∂( x , y , z)
∭ f ( x , y , z ) dV =¿ ∭ f ( T (u , v , w) ) ∂(u , v , w) d u dv dw ¿
T (R ) R