Section 16.2

Line Integrals (Scalar)

Calculating the mass of a wire or the area of a curved fence.

1

Introduction

A Scalar Line Integral sums up values of a function along a curve .

Physical Meaning: If is density, the integral is the Total Mass of the wire C. If is height, it is the Area of the Curtain under along C.

2

Evaluation Formula

Recipe

  1. Parameterize the curve: for .
  2. Compute Magnitude: . This is .
  3. Substitute: Replace in with .
  4. Integrate: .
3

Visualizing the Curtain

Interactive: Area over Semicircle

4

Worked Examples

Example 1: Line Segment

Evaluate where C is the segment from (0,0) to (2,4).

1. Parameterize:

for .

2. Magnitude:

.

.

3. Substitute:

.

.

4. Integrate:


.

Example 2: Mass of a Helix

Find the mass of the wire for if density .

1. Magnitude:

.

.

2. Integral:

.

3. Solve:

.

Example 3: Area of a Cylinder Wall

Find the area of the surface between and .

1. Interpret:

This is a line integral of height along the unit circle C.

2. Parameterize C:

, .

.

3. Integrate:

.

.

5

Practice Quiz

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