1. Tom Wilson, Department of Geology and Geography tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

2. Objectives Tom Wilson, Department of Geology and Geography Continue to develop basic integration skills Use integral concepts to provide information about geological features.

3. To estimate distance covered when velocity varies continuously with time: i.e. v = kt Tom Wilson, Department of Geology and Geography This is an indefinite integral. The result indicates that the starting point is unknown; it can vary. To know where the thing is going to be at a certain time, you have to know where it started. You have to know C.

4. Tom Wilson, Department of Geology and Geography is the area under the curve is the area under a curve, but there are lots of “areas” that when differentiated yield the same v. The instantaneous velocity

5. v=kt Tom Wilson, Department of Geology and Geography The velocity of the object doesn’t depend on the starting point (that could vary)– just on the elapsed time.

6. Tom Wilson, Department of Geology and Geography but - location as a function of time obviously does depend on the starting point.

7. Tom Wilson, Department of Geology and Geography You just add your starting distance ( C ) to That will predict the location accurately after time t

8. Tom Wilson, Department of Geology and Geography Here is an illustration of a definite integral This is referred to as the definite integral and is evaluated as follows

9. Tom Wilson, Department of Geology and Geography & in general … The constants cancel out in this case. You have to have additional observations to determine C or k.

10. At this point, given a starting location, you could make additional computations Tom Wilson, Department of Geology and Geography For example, if you wanted to catch up with that object at a certain time, what would your acceleration have to be to make the intercept? You need to arrive in time t i when the object reaches a distance s i from your location.

11. Tom Wilson, Department of Geology and Geography Before you evaluate this, draw a picture of the cosine and ask yourself what the area will be over this range What is the area under the cosine from  /2 to 3  /2

12. Tom Wilson, Department of Geology and Geography Given where a is a constant; a cannot be a function of x.

13. Tom Wilson, Department of Geology and Geography

14. …and some special rules for x -1 and e kx Tom Wilson, Department of Geology and Geography Compile a list of rules, then go back over the problems we’ve worked and ask yourself what rules you used to do the integrations. Just as with derivatives, there are often different rules that can be employed to achieve the same result.

15. Think about those rules as you do the following (there are a total of 16 problems in the handout). Tom Wilson, Department of Geology and Geography

16. Some geological applications (see Section 9.6) Tom Wilson, Department of Geology and Geography What is the volume of Mt. Fuji? Sum of flat disks

17. Tom Wilson, Department of Geology and Geography riri dz riri is the volume of a disk having radius r and thickness dz. =total volume The sum of all disks with thickness dz Area Radius

18. Tom Wilson, Department of Geology and Geography Waltham notes that for Mt. Fuji, r 2 can be approximated by the following polynomial To find the volume we evaluate the definite integral

19. Tom Wilson, Department of Geology and Geography The “definite” solution

20. Tom Wilson, Department of Geology and Geography We know Mount Fuji is 3,776m (3.78km). So, does the integral underestimate the volume of Mt. Fuji? This is what happens when you carry the calculations on up … It works out pretty good though since the elevation at the foot of Mt. Fuji is about 600-700 meters.

21. See example 9.7 in the text Tom Wilson, Department of Geology and Geography Find the cross sectional area of the sand bar and estimate its volume

22. Let’s get into excel and set it up – get started … Tom Wilson, Department of Geology and Geography Once you get the polynomial, what do you need to do to get the cross sectional area?

23. 4 th order example from the text - Setting up the area computation. Tom Wilson, Department of Geology and Geography t = -2.857E-12x4 + 1.303E-08x3 - 2.173E-05x2 + 1.423E-02x - 7.784E-02

24. We will add some additional precision by using a 5 th order polynomial Tom Wilson, Department of Geology and Geography Note that you will have to increase the precision of the coefficients in your result in order to get an accurate estimate of area.

25. Spend the remainder of the period working on the integrations and computer problem 9.7 Tom Wilson, Department of Geology and Geography What it this integral? Remember you can format trendline and trendline labels …..next time …..

26. Due dates Tom Wilson, Department of Geology and Geography Integral worksheets will be due next Tuesday Bring questions about problem 9.7 to class next Tuesday Finish reading Chapter 9 Look over Question 9.8 and sections 9.7. How do you solve problems 9.9 and 9.10.