Double Integration In Polar Form With Change In Variable

1. Presentation On
“Double Integration In Polar Form
With Change In Variable”
Harsh Gupta
(130800119030)

2. Introductive…
• Let ƒ(r,ө) be the
function of polar
coordinates r and ө,
defined on region R
bounded by the half
lines ө=α, ө=β and
the continuous
curves r=ƒ1(ө),
r=ƒ2(ө), where α ≤ β
and 0 ≤ ƒ1(ө) ≤
ƒ2(ө).Then the region
R is as shown in the
figure.

3. • If a region S in the uv-plane is transformed
into the region R in xy-plane by the
transformation x=ƒ(u,v)=g(x,y) then the
function F(x,y) defined on R can be thought of
a function F(ƒ(u,v),g(u,v)) defined on S.

4. • ∫∫R ƒ(r,ө) dA = ∫∫R ƒ(r,ө) r dr dө

5. • In this type of examples x is replaced by
(r cosө) and y is replaced by (r sinө ).
And dx and dy are replaced by (r dr dө).
Here r comes from Jacobian solution of
x= r cosө and y= r sinө .
And x² + y² = r²..
i.e. : x-> r cosө
y-> r sinө

6. Some Examples…

9. Real Life Examples..
• Velocity Profile with No Slip Boundary Condition
• Double Integrals can be used to calculate the Mass Flow of air
entering the Airbox of an F1 car by using Polar Coordinates
and Velocity Profiles. This is a useful real world application for
double integrals, and there are many others. In the near
future I hope to write a follow up with comparisons to CFD
and Hagen-Poiseuille Flow.
• - See more at: http://consultkeithyoung.com/content/applied-
math-example-double-integral-polar-coordinates-flow-f1-
airbox-inlet#sthash.OuIwgNjv.dpuf

13. • Determine the volume of the region that lies
under the sphere , above the plane and inside
the cylinder .
Solution
• We know that the formula for finding the
volume of a region is,
• In order to make use of this formula we’re
going to need to determine the function that
we should be integrating and the region D that
we’re going to be integrating over.

14. • The function isn’t too bad. It’s just the sphere, however, we
do need it to be in the form . We are looking at the region
that lies under the sphere and above the plane
• (just the xy-plane right?) and so all we need to do is solve the
equation for z and when taking the square root we’ll take the
positive one since we are wanting the region above the xy-
plane. Here is the function.
• The region D isn’t too bad in this case either. As we take
points, , from the region we need to completely graph the
portion of the sphere that we are working with. Since we only
want the portion of the sphere that actually lies inside the
cylinder given by this is also the region D. The region D is the
disk in the xy-plane.
• For reference purposes here is a sketch of the region that we
are trying to find the volume of.

15. • So, the region that we want the volume for is
really a cylinder with a cap that comes from
the sphere.
• We are definitely going to want to do this
integral in terms of polar coordinates so here
are the limits
(in polar coordinates) for the
region,
• and we’ll need to convert the
function to polar coordinates
as well.
• The volume is then,