5.1 Estimating with Finite Sums

Greenfield Village, Michigan

time

velocity

After 4 seconds, the object has gone 12 feet.

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.

ft3 4 sec 12 ft

sec

3t d

When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.

211

8V t

subinterval

partition

The width of a rectangle is called a subinterval.

The entire interval is called the partition.

Subintervals do not all have to be the same sizebut for today they will.

If the velocity is not constant,we might guess that the distance traveled is still equalto the area under the curve.

(The units work out.)

211

8V t 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).

11

18

11

2

12

8

t v

10

1 11

8

2 11

2

3 12

8Approximate area: 1 1 1 3

1 1 1 2 5 5.758 2 8 4

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

11

8

11

2

12

8

Approximate area: 1 1 1 31 1 2 3 7 7.75

8 2 8 4

3

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

1.031251.28125

1.78125

Approximate area:6.625

2.53125

t v

1.031250.5

1.5 1.28125

2.5 1.78125

3.5 2.53125

In this example there are four subintervals.As the number of subintervals increases, so does the accuracy.

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8V t

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Approximate area:6.65624

t v

1.007810.25

0.75 1.07031

1.25 1.19531

1.382811.75

2.25

2.75

3.25

3.75

1.63281

1.94531

2.32031

2.75781

13.31248 0.5 6.65624

width of subinterval

With 8 subintervals:

The exact answer for thisproblem is .6.6

Circumscribed rectangles are all above the curve:

Inscribed rectangles are all below the curve:

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.

If you have the calculus tools programinstalled:

Set up the WINDOW screen as follows:

Select Calculus Tools and press Enter

Press APPS

Press F3

Press alpha and then enter: 1/ 8 ^ 2 1x

Make the Lower bound: 0Make the Upper bound: 4Make the Number of intervals: 4

Press Enter

and then 1

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

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