Triple Integrals in Cylindrical Coordinates: Fall 2009, UPenn, Tong Zhu, Study notes of Mathematics

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Math 114-004, Fall 2009

Tong Zhu

Department of Mathematics University of Pennsylvania

December 1, 2009

Triple Integrals in Cylindrical Coordinates

Triple Integrals in Cylindrical Coordinates

Recall: polar coordinate

x = r cos θ y = r sin θ

Triple Integrals in Cylindrical Coordinates

Recall: polar coordinate

x = r cos θ y = r sin θ

r 2 = x^2 + y 2 tan θ =

y x

Cylindrical Coordinates

In three dimensional space, a point P can be represented by the ordered triple (r , θ, z):

Cylindrical Coordinates

In three dimensional space, a point P can be represented by the ordered triple (r , θ, z):

r 2 = x^2 + y 2 tan θ =

y x

z = z

Example 1

(^1) Plot the point whose cylindrical coordinate is (2, π/ 4 , 1). (^2) Identify the surface z = 4 − r 2.

Triple Integrals in Cylindrical Coordinates

Consider the “Type 1” region: E = {(x, y , z)|(x, y ) ∈ D, u 1 (x, y ) ≤ z ≤ u 2 (x, y )}

Triple Integrals in Cylindrical Coordinates

Consider the “Type 1” region: E = {(x, y , z)|(x, y ) ∈ D, u 1 (x, y ) ≤ z ≤ u 2 (x, y )}

In this case, the triple integral can be written as ∫ ∫ ∫

E

f (x, y , z)dV =

D

[ ∫^ u 2 (x,y )

u 1 (x,y )

f (x, y , z)dz

]

dA

Triple Integrals in Cylindrical Coordinates

Consider the “Type 1” region: E = {(x, y , z)|(x, y ) ∈ D, u 1 (x, y ) ≤ z ≤ u 2 (x, y )}

In this case, the triple integral can be written as ∫ ∫ ∫

E

f (x, y , z)dV =

D

[ ∫^ u 2 (x,y )

u 1 (x,y )

f (x, y , z)dz

]

dA

Suppose D can be given by polar coordinates: D = {(r , θ)|α ≤ θ ≤ β, h 1 (θ) ≤ r ≤ h 2 (θ)}

∫ ∫ ∫

E

f (x, y , z)dV

∫ (^) β

α

∫ (^) h 2 (θ)

h 1 (θ)

∫ (^) u 2 (r cos θ,r sin θ)

u 1 (r cos θ,r sin θ)

f (r cos θ, r sin θ, z)rdzdrdθ

Example 2 Evaluate

− 2

∫ √ 4 −y 2

−

4 −y 2

x^2 +y 2

xzdzdxdy.