Section 15.6
Cylindrical Coordinates
When the domain is circular but the solid has height.
1
Introduction
Cylindrical coordinates are just Polar Coordinates in the xy-plane, plus a z-axis.
2
Integration Formula
Volume Element
Don't forget the Jacobian !
3
Visualizing Cylinders
Interactive: Paraboloid Volume
4
Worked Examples
Example 1: Mass of a Cylinder
Find mass of cylinder if density .
1. Limits:
.
.
.
2. Integral:
.
3. Solve:
z-integral: .
r-integral: .
theta-integral: .
Example 2: Volume of Paraboloid
Find volume bounded by and .
1. Setup:
.
Intersection w/ floor: .
.
2. Execute:
z-int: .
r-int: .
theta-int: .
Example 3: Cone Volume
Find volume bounded by cone and plane .
1. Bounds:
Cone is .
Top is .
Intersect at .
2. Integrate:
(Floor is cone, ceiling is plane).
3. Solve:
z-int: .
r-int: .
theta-int: .
5
Practice Quiz
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