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Geomath

Geomath. Geology 351 -. Integral Calculus Continued. tom.h.wilson tom.wilson@mail.wvu.edu. Dept. Geology and Geography West Virginia University. Recall how we estimate distance covered when velocity varies continuously with time: v = kt. This is an indefinite integral.

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Geomath

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  1. Geomath Geology 351 - Integral Calculus Continued tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University Tom Wilson, Department of Geology and Geography

  2. Recall how we estimate distance covered when velocity varies continuously with time: v = kt 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. Tom Wilson, Department of Geology and Geography

  3. The instantaneous velocity 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. Tom Wilson, Department of Geology and Geography

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

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

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

  7. 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 Tom Wilson, Department of Geology and Geography

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

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

  10. Some elementary integration rules “Constant factor" rule for integrals Given where a is a constant; a cannot be a function of x. Tom Wilson, Department of Geology and Geography

  11. Integral of a sum Power rule for integrals Any Questions? Tom Wilson, Department of Geology and Geography

  12. Volumes too ... What is the volume of Mt. Fuji? Sum of flat disks Tom Wilson, Department of Geology and Geography

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

  14. Waltham notes that for Mt. Fuji, r2 can be approximated by the following polynomial To find the volume we evaluate the definite integral Tom Wilson, Department of Geology and Geography

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

  16. Problem 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. Tom Wilson, Department of Geology and Geography

  17. Conduction equation 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. Tom Wilson, Department of Geology and Geography

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

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

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

  21. The Fourier series can be expressed more compactly using summation notation Fourier series Bring upStep.xls 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.

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

  26. Set-up Tom Wilson, Department of Geology and Geography

  27. 4th order example from the text t = -2.857E-12x4 + 1.303E-08x3 - 2.173E-05x2 + 1.423E-02x - 7.784E-02 Tom Wilson, Department of Geology and Geography

  28. 1st let's work through this one... in-class integrations for additional practice Work in groups 1-4 Tom Wilson, Department of Geology and Geography

  29. Due dates • Hand integral worksheets in before you leave today • Bring questions in about problem 9.7 next Tuesday. Tom Wilson, Department of Geology and Geography

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