Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives: ...
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Transcript of Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives: ...
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Chapter 15 – Multiple Integrals15.4 Double Integrals in Polar Coordinates
15.4 Double Integrals in Polar Coordinates
Objectives: Determine how to express
double integrals in polar coordinates
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 2
Polar Coordinates ReviewRecall from this figure that the polar coordinates (r, θ) of
a point are related to the rectangular coordinates (x, y) by the equations◦ r2 = x2 + y2
◦ x = r cos θ◦ y = r sin θ
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 3
Double Integrals in Polar Coordinates
Suppose that we want to evaluate a double integral
, where R is one of the regions shown
here.
( , )R
f x y dA
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 4
Double Integrals in Polar Coordinates
In either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates.
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 5
Polar RectangleThe regions in the first figure are special cases of a polar
rectangle R = {(r, ) | a ≤ r ≤ b, ≤ ≤ }
shown here.
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 6
Polar RectangleTo compute the double integral
where R is a polar rectangle, we divide:
◦ The interval [a, b] into m subintervals [ri–1, ri] of equal width ∆r = (b – a)/m.
◦ The interval [ ,] into n subintervals [ j–1, j] of equal width ∆ = ( – )/n.
( , )R
f x y dA
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 7
Polar SubrectangleThe “center” of the polar subrectangle
Rij = {(r, θ) | ri–1 ≤ r ≤ ri, θj–1 ≤ θ ≤ θi}
has polar coordinates
ri* = ½ (ri–1 + ri)
θj* = ½ (θj–1 + θj)
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 8
Polar RectanglesThe rectangular coordinates of the center of Rij are (ri*
cos θj*, ri* sin θj*).
So, a typical Riemann sum is shown with Equation 1:
* * * *
1 1
* * * * *
1 1
( cos , sin )
( cos , sin )
m n
i j i j ii j
m n
i j i j ii j
f r r A
f r r r r
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 9
Polar Rectangles If we write g(r, θ) = r f(r cos θ, r sin θ), the Riemann
sum in Equation 1 can be written as:
◦ This is a Riemann sum for the double integral
* *
1 1
( , )m n
i ji j
g r r
( , )b
ag r dr d
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 10
Polar RectanglesThus, we have:
* * * *
,1 1
* *
,1 1
( , ) lim ( cos , sin )
lim ( , )
( , )
( cos , sin )
m n
i j i j im n
i jR
m n
i jm n
i j
b
a
b
a
V f x y dA f r r A
g r r
g r dr d
f r r r dr d
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 11
Change to Polar Coordinates
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 12
Change to Polar CoordinatesFormula 2 says that we convert from rectangular to
polar coordinates in a double integral by:
◦Writing x = r cos θ and y = r sin θ◦Using the appropriate limits of integration
for r and θ◦Replacing dA by dr dθ
Be careful not to forget the additional factor r on the right side of Formula 2.
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 13
Change to Polar CoordinatesA classical method for remembering
the formula is shown here.
◦ The “infinitesimal” polar rectangle can be thought of as an ordinary rectangle with dimensions r dθ and dr.
◦ So, it has “area” dA = r dr dθ.
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 14
Example 1Sketch the region whose area is given by
the integral and evaluate the integral.
4cos2
0 0
rdrd
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 15
Example 2Evaluate the integral by changing to polar
coordinates.
2 2 2 2
, where is the region in the first quadrant that
lies between the circles 4 and 2 .
D
xdA D
x y x y x
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 16
More Complicated RegionsWhat we have done so far can be extended to the more
complicated type of region shown here.
◦ It’s similar to the type II rectangular regions considered in Section 15.3
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 17
More Complicated Volumes If f is continuous on a polar region
of the form
D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}
then2
1
( )
( )( , ) ( cos , sin )
h
hD
f x y dA f r r r dr d
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 18
Volumes and Areas In particular, taking f(x, y) = 1, h1(θ) = 0,
and h2(θ) = h(θ) in the formula, we see that the area of the region D bounded by θ = α, θ = β, and r = h(θ) is:
Which agrees with formula 3 in section 10.4.
( )
0
( )221
2
0
( ) 1
[ ( )]2
h
D
h
A D dA r dr d
rd h d
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 19
Example 3Use a double integral to find the area of
the region.
The region inside the cardiod 1 cos and outside
the circle 3cos .
r
r
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 20
Example 4 Use polar coordinates to find the volume
of the given solid.2 2Bounded by the paraboloid 1 2 2
and the plane 7 in the first octant.
z x y
z
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 21
Example 5 Evaluate the iterated integral by
converting to polar coordinates.22 2
2 2
0 0
x x
x y dydx
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 22
More Examples
The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 2◦Example 3◦Example 4
Dr. Erickson
15.4 Double Integrals in Polar Coordinates 23
DemonstrationsFeel free to explore these demonstrations
below.◦Double Integral for Volume
Dr. Erickson