MATH 1351 TI-85 EXERCISE XIII

Area estimations with the TI-85

Name: SID:

The idea is to use sums of left boxes and sums of right boxes to estimate the area under the graph of a positive function y = f ( x ) , between the values x = a and x = b. We recall the syntax:

left box approximation = sum seq( delta-x f ( a + I delta-x ) , I , 0 , N-1 , 1)

right box approximation = sum seq( delta-x f ( a + I delta-x ) , I , 1 , N , 1) .

Exercise 1. Attach a picture that illustrates the validity of the following statements.

A. If the function f ( x ) is increasing on the interval [ a , b ], then

lt box appx < actual area < rt box appx .

B. If the function f ( x ) is decreasing on the interval [ a , b ], then

rt box appx < actual area < lt box appx.

Thus the possible error in using either approximation is less than | rt box appx - lt box appx |.

Exercise 2. Use these ideas to estimate the area under the graph of y = 1/x , between x = 1 and x = 3. (Compute the lt and rt box appx's until their difference is less than .005.) Record your data in the table provided.

# of boxes	lt box appx	rt box appx	possible error

Final estimate of the area ( rounded off to 2 decimal places ) is

Exercise 3. Estimate the area under the graph of y = e^(x^2) , between x = 0 and x = 2. Again record your data in the table provided.

# of boxes	lt box appx	rt box appx	possible error

Final estimate of the area ( rounded off to 2 decimal places ) is

Exercise 3. Suppose you know that some number is between 1 and 5, but that's all you know about it. What would be your "best" guess as to the number? (That is, knowing only that the number is between 1 and 5, what guess would be guaranteed to minimize the possible error?)

We know that the actual area (for an increasing or decreasing, monotone, function ) is between lt box appx and rt box appx. So the "obvious" best guess is half way between them. This will be called the trapezoidal approximation:

trap appx = ( lt box appx + rt box appx ) / 2 .

Exercise 2' . From the above data, use the trapezoidal rule to approximate the area under the graph of y = 1 / x, between x = 1 and x = 3. What is the fewest number of "boxes" required to make the trap appx within 2 decimal places of the answer you obtained previously?

Exercise 3' . From the above data, use the trapezoidal rule to approximate the area under the graph of y = e^(x^2), between x = 0 and x = 2. What is the fewest number of "boxes" required to make the trap appx within 2 decimal places of the answer you obtained previously?

Exercise 4. Compare each of your above approximations with the answer obtained by using the TI-85 numerical integrator directly. It is accessed by choosing 2nd CALC from the keyboard then fnInt from the screen menu. The syntax is

fnInt( f(x) , x , a , b ) ,

read "the definite integral of f (x) , with respect to x form x = a to x = b."

Exercise 5. Why do you think the above rule is called the trapezoidal rule?