Section 15.9
Surface Area
Extending arc length to 3D.
1
Introduction
To find the area of a curved surface , we approximate it with tangent planes.
A small rectangle on the floor corresponds to a tilted parallelogram on the surface.
2
The Surface Area Formula
Definition
For a surface over a region D:
This factor represents the "tilt" of the surface.
3
Visualizing Surface Area
Interactive: Paraboloid Cap
4
Worked Examples
Example 1: Surface Area of Paraboloid
Find the area of below .
1. Partial Derivatives:
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2. Convert to Polar:
Region is circle (radius 3).
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3. Solve:
u-sub: .
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Limits: , .
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Example 2: Plane in First Octant
Find area of in the first octant.
1. Derivatives:
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2. Area of D:
D is the triangle in xy-plane where , i.e., .
Intercepts: (3,0) and (0,2). Area = .
3. Surface Area:
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Example 3: Sphere Surface Area
Find the area of .
1. Upper Hemisphere:
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2. Integral:
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Sub .
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3. Total Area:
Double it for lower hemisphere: .
5
Practice Quiz
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