Surface Integrals
Extending double integrals to curved surfaces in 3D space.
Introduction
A Surface Integral calculates the total accumulation of a scalar quantity (like mass, charge, or heat) distributed over a surface .
If , the integral gives the Surface Area of S.
Evaluation Formula
Recipe
To evaluate, we parameterize the surface over a domain D in the -plane. The area element becomes:
Full Integral:
Visualizing Weighted Area
Interactive: Density on a Sphere
Integrating f over the surface sums up the weighted contribution of each patch.
Worked Examples
Example 1: Mass of a Plane Segment
Find the mass of the surface over the square with density .
1. Parameterize (Monge Patch):
.
, .
2. Cross Product Magnitude:
.
.
3. Integrate:
.
.
.
Example 2: Surface Area of Paraboloid
Find area of below .
1. Use Polar Domain:
Domain D is disk (radius 2).
.
.
2. Integral:
.
3. Solve (u-sub):
Let .
.
Example 3: Sphere Mass
Find mass of sphere if density is .
1. Parameterize:
Use Spherical. .
Density .
2. Integral:
.
Separates into: .
3. Result:
.
Practice Quiz
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