5.1 Estimating with Finite Sums

320 miles

50 ft/sec

12,000 gallons

-3

0

1

2

3

1 2 3 4

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

0

1

2

3

1 2 3 4

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).

1 11

8

11

2

12

8t 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

0

1

2

3

1 2 3 4

211

8V t

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 2.531250

1

2

3

1 2 3 4

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

211

8V t

211

8V t

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

0

1

2

3

1 2 3 4

With 8 subintervals:

The exact answer for thisproblem is .6.6

Circumscribed rectangles are all above the curve:

0

1

2

3

1 2 3 4

0

1

2

3

1 2 3 4

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.