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

2. Today’s agenda Tom Wilson, Department of Geology and Geography April madness/Integration boot camp: let’s spend some additional time on the basics We’ll also Fault propagation fold – area computation and shortening estimation (hand in before leaving).

3. Refer to handout from last time Tom Wilson, Department of Geology and Geography Detachment horizon Volume of detached rock forced into the fold We approximate the shape of the deeper fold as We approximate the shape of the shallower detached fold as -x+x

4. Fold relief is approximated by two quadratic equations Tom Wilson, Department of Geology and Geography Note that the limits used here coincide with the area for which the blue curve is greater than the brown curve (-5kft to 5kft)

5. A structural geology problem cast in terms of calculus concepts Tom Wilson, Department of Geology and Geography Detachment horizon What is the excess area in this cross sectional view?

6. Calculate the area between these two curves Tom Wilson, Department of Geology and Geography Evaluate This is referred to as a definite integral. The area (or difference of areas in this case) is computed only over a certain limited range corresponding to the extent of the shallow detached fold. Finish up this problem and turn in before leaving today

7. Take a few minutes to evaluate the integral and estimate the shortening Tom Wilson, Department of Geology and Geography

8. Recall volume integration approach from last time 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

9. We ended up with the definite integral Tom Wilson, Department of Geology and Geography

10. For the sand bar problem you are going to create a 5 th order polynomial and integrate it. Tom Wilson, Department of Geology and Geography The 5 th order polynomial you derive in Excel will give you 6 terms including the constant What is this integral? But you’ll let Excel do the heavy lifting on the calculations

11. What kinds of questions do you have on problem 9.7? Tom Wilson, Department of Geology and Geography Find the cross sectional area of the sand bar and estimate its volume 1 st Integrate the polynomial representation of sand bar cross section to get the area.

12. The Excel computation is illustrated using the 4 th order polynomial presented by Waltham Tom Wilson, Department of Geology and Geography t = -2.857E-12x 4 + 1.303E-08x 3 - 2.173E-05x 2 + 1.423E-02x - 7.784E-02 The constants a 4 a2a2 a3a3 a1a1 a0a0 You have to integrate the polynomial to get the area. A possible Excel set up is shown below.

13. For the homework, 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.

14. Let’s take a few minutes and dig into this problem and make sure you have it set up properly Tom Wilson, Department of Geology and Geography Remember you can format trendline and trendline labels

15. Due dates Tom Wilson, Department of Geology and Geography Fault propagation fold – area computation and shortening estimation (hand in before leaving). The computer version of Question 9.7 will be due on the 17th. If time permits > Problems 9.9 and 9.10 – Questions? Definitely bring those questions to class on Thursday 9.9 and 9.10 are tentatively due on the 22 nd (next Tuesday)to be returned and discussed on the 24 th.