1. Triple Integral in Cylindrical Coordinates

2. Triple Integrals in Cylindrical coordinates
Cylindrical coordinates of the point P: (𝑟,𝜃,𝑧)
𝑟 and 𝜃 are the polar coordinates of the projection of the point P onto the 𝑥𝑦-plane.
𝑧 is the signed vertical distance between P and the 𝑥𝑦-plane (same as in cartesian)
From Cylindrical to Cartesian:
From Cartesian to Cylindrical:

3. Triple Integrals in Cylindrical coordinates
Example 1: Given find x, y and z.
Conversely, given rectangular (−3,−3,7) find cylindrical:
is a possible answer

4. Triple Integrals in Cylindrical Coordinates
Basic graphs in cylindrical coordinates:
𝑟=𝑐 represents a cylinder
( 𝑥 2 + 𝑦 2 = 𝑐 2 in cartesian)
𝜃=𝑐 represents a vertical plane (if
r ≥ 0, half a plane)
𝑧=𝑐 represents a horizontal plane
𝑧=𝑟 represents the cone
𝑧= 𝑥 2 + 𝑦 2

5. Triple Integrals in cylindrical coordinates
Since 𝑥 2 + 𝑦 2 = 𝑟 2 , integrals involving 𝑥 2 + 𝑦 2 or
𝑥 2 + 𝑦 2 frequently are easier in cylindrical coordinates.
The volume element is
Theorem: (Change of coordinates)
Let E be the region:
Then the triple integral of f over E in cylindrical coordinates is

6. Triple Integrals in cylindrical coordinates – Example 2
Evaluate where E lies above z = 0, below z = y and inside the cylinder 𝑥 2 + 𝑦 2 =9.
The plane 𝑧=𝑦 in cylindrical coordinates is 𝑧=𝑟 sin 𝜃
The domain D is the semicircle of radius 3: 0≤𝑟≤3, 0≤𝜃≤𝜋
The integrand function 𝑦𝑧 in cylindrical coordinates is (𝑟 sin 𝜃)𝑧

7. Triple Integrals in Cylindrical Coordinates – Example 3
Find the volume of the cone 𝑧= 𝑥 2 + 𝑦 for 𝑧≤4 using cylindrical coordinates.
The cone and the plane 𝑧=4 intersect in a circle:
This circle defines the boundaries for 𝑟 and 𝜃:
0≤𝑟≤4, 0≤𝜃≤2𝜋
In cylindrical coordinates the cone has equation 𝑧=𝑟, thus 𝑟≤𝑧≤4

8. Triple Integrals in Cylindrical Coordinates – Example 4
Sketch the solid whose volume is given by the integral and evaluate the integral.
The solid is bounded below by 𝑧=0 (the 𝑥𝑦-plane) and above by the paraboloid 𝑧=9− 𝑟 2 =9− 𝑥 2 − 𝑦 2
The solid is bounded by the cylinder 𝑟=2
( in cartesian: 𝑥 2 + 𝑦 2 =4)
The solid is in the first octant (𝑥≥0 and 𝑦≥0)