## 4.1 Modeling Accumulations

### Summary

The issues to be studied in Chapter 4, Modeling Accumulations, are introduced. In particular, the problem of finding the area under a curve and above an interval on the x-axis is described. The method of accumulations is also illustrated; several examples are given. An agenda is established for the rest of the chapter.

By the end of your studying, you should know:

• How to approximate the area of a circle by beginning with a hexagon and doubling the number of sides repeatedly in succession.
• The subtlety of defining the area of a circle.
• How to approximate the area of a quarter-circle with rectangles.
• How to describe the method of accumulations.
• How to use the method of accumulations to approximate the circumference of a circle.
• How to use the method of accumulations and rainfall data to approximate the volume of water that flows past a fixed point in a stream in the hours immediately following a rainstorm.

On-screen applet instructions: Use the pull-down menu to choose the number of subintervals and hence rectangles. A mouse click on the figure will toggle between a close view and a wider view of the circle.

### Examples

Approximate the area of an ellipse using 7 circumscribed rectangles in the first quadrant.

Approximate the area of the quarter circle of radius 3 centered at the origin using n circumscribed rectangles, giving your answer in terms of n.

The graph of a stream's rate of flow over a 1-hour period is the graph of a sine curve. Assuming the rate of flow is measured every 10 minutes, starting at time 0, use the method of accumulation to aproximate the total volume of flow over this hour, accurate to 3 decimal places.

### Applets

Approximating Areas: Inscribed Polygons

Approximating Area: Using Rectangles

Accumulation: River Flow

Accumulation: Distance Traveled

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 4.1 Modeling Accumulations (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

History of Calculus
O'Connor and Robertson

Approximation of Areas
Visual Calculus

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel