Using Your Calculator to Integrate

Problem Set 18 - Using your calculator to integrate

Example 1: (The following example is taken from Calculus, by James Stewart, 5th edition)

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 seconds. The maximum rate of air flow into the lungs is about .5 L/s.

a. Graph the function f (t)  .5sin(2 t / 5) on the interval [0,20] using your calculator and sketch it below. Be sure that your calculator is in radian mode. Explain why it is reasonable to use this function to model the rate of air flow in the lungs. (You may need to adjust your viewing rectangle to see the graph.)

We will now use this model to find the volume of inhaled air in the lungs after 12 seconds. 12 b. Explain why the integral  f (t)dt represents the volume of inhaled air in the lungs after 12 0 seconds and show this graphically on your sketch in part a.

c. Give a rough estimate for the volume of air after 12 seconds, based on your knowledge of geometry.

d. Now we’ll use the calculator to compute it. (Refer to the Calculus with the TI-84 handout, “Computing a definite integral” from the section “From the graph of a function”.) Compare the result with your estimate.

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1 Problem Set 18 - Using your calculator to integrate

2 Example 2 (no calculator): We wish to determine the area bounded by the two curves y1  x 2 y2  2x  x .

a. First sketch the two functions on the interval [-2,2].

b. Shade the area that is bounded by the two functions.

c. Determine where the two graphs intersect so that you know the boundaries for the shaded region.

d. Just as we did before, we could approximate this area by using rectangles. Sketch one such rectangle that could be used to approximate a portion of this area. What is the height of the rectangle you sketched?

e. Based on how you answered part d., write the integral for the area bounded by these two curves.

f. Evaluate the integral you wrote in e, and compute this integral using your calculator to check it.

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 2 Problem Set 18 - Using your calculator to integrate

Example 4: Using your calculator, shade the area of the region bounded by the curves x 4 y1  and y  x  x , and then compute this area. x 2 1 2

Suggestion: To find where the two curves intersect, first graph both in an appropriate viewing rectangle. Then press the TRACE button to get close to a point of intersection. Now press 2ND CALCULATE and type 5 to access the INTERSECT command. Press ENTER three times, and the approximate point of intersection will be displayed.

Copyright 2007. Concepts of Calculus for Middle Level Teachers. First developed for the La Meta Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 3