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Photo by Vickie Kelly, 2002. Greg Kelly, Hanford High School, Richland, Washington. 5.1 Estimating with Finite Sums. Greenfield Village, Michigan. velocity. time. Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance:.

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Presentation Transcript
slide1

Photo by Vickie Kelly, 2002

Greg Kelly, Hanford High School, Richland, Washington

5.1 Estimating with Finite Sums

Greenfield Village, Michigan

slide2

velocity

time

Consider an object moving at a constant rate of 3 ft/sec.

Since rate . time = distance:

If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

After 4 seconds, the object has gone 12 feet.

slide3

Approximate area:

If the velocity is not constant,

we might guess that the

distance traveled is still equal

to the area under the curve.

(The units work out.)

Example:

We could estimate the area under the curve by drawing rectangles touching at their left corners.

This is called the Left-hand Rectangular Approximation Method (LRAM).

slide4

Approximate area:

We could also use a Right-hand Rectangular Approximation Method (RRAM).

slide5

Approximate area:

Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).

In this example there are four subintervals.

As the number of subintervals increases, so does the accuracy.

slide6

Approximate area:

The exact answer for this

problem is .

With 8 subintervals:

width of subinterval

slide7

Circumscribed rectangles are all above the curve:

Inscribed rectangles are all below the curve:

slide8

We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation.

The TI-89 calculator can do these rectangular approximation problems. This is of limited usefulness, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.

slide9

Set up the WINDOW screen as follows:

If you have the calculus tools program

installed:

slide10

Press APPS

Select Calculus Tools and press Enter

Press F3

and then 1

Press alpha and then enter:

Press Enter

Note: We press alpha because the screen starts in alpha lock.

Make the Lower bound: 0

Make the Upper bound: 4

Make the Number of intervals: 4