Section 6.2

Area Between Curves

The area under one curve is . The area between two curves is the space trapped in the middle. We define it as "Height * Width".

1

Scanning the Area


Always subtract the bottom function from the top to ensure positive height.

The Area Scanner

Scan Area = Sum of heights $(Top - Bottom)dx$

Notice: The height of the strip is always determined by Blue minus Red.
2

Worked Example

Intersecting Curves

Find area between and .

1. Find Intersections


Roots: .
2. Identify Top/Bottom

Test :

  • Top:
  • Bottom:
3. Integrate

.
3

Level Up Examples

Example A: Sine vs Cosine

Area between and on .

1. Top:
Cosine starts at 1, Sine at 0. So .
2. Integrate:


.

Example B: Integrating w.r.t y

Area between and .

1. Bounds (y):
.
2. Right - Left:
The Line () is to the right of Parabola.
3. Integrate:
.
4

Integrating with Respect to y

Sometimes curves are easier to describe as functions of . In this case, we integrate horizontally:

Horizontal Area Scanner

Area = Sum of widths

y = 0.50 → Width = 2.25
The width of each horizontal strip is Right () minus Left ().

Example: Parabola and Line

Find area between and .

1. Find Intersections (solve for y)


Roots: .
2. Identify Right/Left

Test :

  • Right: (the line)
  • Left: (the parabola)
3. Integrate



Example: Two Sideways Parabolas

Find area between and .

1. Intersections:
.
2. Right - Left:
At : Right is , Left is .
3. Integrate:

.
5

Practice Quiz

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