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

2. Recall how we estimate distance covered when velocity varies continuously with time: 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.

3. 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

4. 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.

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

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

7. Tom Wilson, Department of Geology and Geography There’s another class of integrals in which the limits of integration are specified, such as This is referred to as the definite integral and is evaluated as follows

8. 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.

9. 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

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

11. Tom Wilson, Department of Geology and Geography

12. What is the volume of Mt. Fuji? Sum of flat disks

13. 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

14. 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

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

16. 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.

17. Conduction equation Tom Wilson, Department of Geology and Geography What is the heat flow from the sill below? X=0 km X=40 km T is given as Assuming that temperature (T) is in centigrade, what are the units of the constants in this equation.

18. Calculate the temperature gradient Tom Wilson, Department of Geology and Geography Given K and that 1 heat flow unit = Calculate q x at x=0 and 40km.

19. 5*sin (2  4t) Amplitude = 5 Frequency = 4 Hz seconds Fourier said that any single valued function could be reproduced as a sum of sines and cosines Introduction to Fourier series and Fourier transforms

20. Fourier series: a weighted sum of sines and cosines Periodic functions and signals may be expanded into a series of sine and cosine functions

21. Fourier series The Fourier series can be expressed more compactly using summation notation You’ve seen from the forgoing example that right angle turns, drops, increases in the value of a function can be simulated using the curvaceous sinusoids. Bring upStep.xls

22. Fourier series

23. This applet is fun to play with & educational too. Experiment with http://www.falstad.com/fourier/

24. You can also observe how filtering of a broadband waveform will change audible waveform properties. http://www.falstad.com/dfilter/

25. Tom Wilson, Department of Geology and Geography Example 9.7 - find the cross sectional area of a sedimentary deposit (see handout).

26. Tom Wilson, Department of Geology and Geography

27. t = -2.857E-12x4 + 1.303E-08x3 - 2.173E-05x2 + 1.423E-02x - 7.784E-02

28. Tom Wilson, Department of Geology and Geography

29. 1.Hand integral worksheets in before you leave today 2.Bring questions in about problem 9.7 next Tuesday.