Section 5.5

The Area Problem

How do we measure the area of a shape with curved edges? The Greeks exhausted the shape with polygons. We use rectangles—lots of them.

1

Riemann Sums

We approximate area by summing the areas of rectangles.

Riemann Sums

Approximating Area under

Approximation
0.0000
Exact Area
2.6667
Error
2.6667
2

Worked Example: Right Endpoints

Manual Estimation

Estimate area under on using 4 rectangles and Right Endpoints.

1. Find Width ()
2. Find Endpoints

Start at 0, add 0.5 each time.

  • 0.5
  • 1.0
  • 1.5
  • 2.0
3. Calculate Sum


3

Level Up Examples

Example A: Left Endpoints

Same problem ( on , ) but using Left Endpoints.

1. Endpoints:
(We stop before the last edge).
2. Sum:

.

Notice Right (3.75) was too big, Left (1.75) is too small.

Example B: Midpoints (More Accurate)

Function on with Midpoints.

1. Width:
.
2. Midpoints:
Intervals and .
Mids: and .
3. Sum:
.
5

Practice Quiz

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