MATH2310 – MVC Lecture 4
Annuli:
An annulus or polar rectangle is the region:
R=[ r 1 , r 2 ] × [ θ1 , θ2 ]
Riemann sums over annuli:
Annuli can be divided into sub annuli.
The area of a single annulus is given by:
θ 2−θ1 2
A=
2
( r 2 −r 12 )
This can be re-written as:
¿
A=r ∆ r ∆ θ
If the region has been subdivided into nm sub-annuli, then:
r 2−r 1 θ2−θ1
∆r= , ∆θ=
n m
The volume of the sample prison at point ( r ¿ ,θ¿ ) is:
¿ ¿ ¿
r ∆ r ∆ θ ∙ f (r , θ )
Polar double integrals:
For a real polar function with R=[ r 1 , r 2 ] ×[θ 1 , θ2 ]:
❑ θ2 r2 n m
∬ g ( r ,θ ) dA=∫∫ g ( r , θ ) r dr dθ= n,lim
m→ ∞
∑ ∑ g(r i¿ , ¿ θ j¿ ) r ¿ ∆ r ∆θ ¿
R θ1 r1 i=1 j=1
Annuli:
An annulus or polar rectangle is the region:
R=[ r 1 , r 2 ] × [ θ1 , θ2 ]
Riemann sums over annuli:
Annuli can be divided into sub annuli.
The area of a single annulus is given by:
θ 2−θ1 2
A=
2
( r 2 −r 12 )
This can be re-written as:
¿
A=r ∆ r ∆ θ
If the region has been subdivided into nm sub-annuli, then:
r 2−r 1 θ2−θ1
∆r= , ∆θ=
n m
The volume of the sample prison at point ( r ¿ ,θ¿ ) is:
¿ ¿ ¿
r ∆ r ∆ θ ∙ f (r , θ )
Polar double integrals:
For a real polar function with R=[ r 1 , r 2 ] ×[θ 1 , θ2 ]:
❑ θ2 r2 n m
∬ g ( r ,θ ) dA=∫∫ g ( r , θ ) r dr dθ= n,lim
m→ ∞
∑ ∑ g(r i¿ , ¿ θ j¿ ) r ¿ ∆ r ∆θ ¿
R θ1 r1 i=1 j=1
