Tom Wilson, Department of Geology and Geography tom.h.wilson Dept. Geology and Geography West Virginia University

Due dates Tom Wilson, Department of Geology and Geography Hand in computer problems 8.13 and 8.14 before leaving today Finish reading chapter 9 and look over the problems in the text discussed in the applications section 9.6. How would you approach problems 9.7 and 9.8 Also look over questions 9.9 and They will be assigned … down the road Spend some more time working through those derivatives in reverse

Objectives for the day Tom Wilson, Department of Geology and Geography Tie up loose ends on derivatives and derivative applications discussed in Chapter 8 Introduce integral concepts. Evaluating the area covered by a function. Use Waltham excel files to illustrate integral relationships Distinguish between the indefinite and definite integrals Continue to sharpen integration and problem solving skills

We introduced the notion of the antiderivative Tom Wilson, Department of Geology and Geography What is it that we have to differentiate to get the functions at left? We’ll come back to these later. Tuesday’s worksheet should be handed in before leaving today.

Looking at the derivatives of functions and guessing the original function Tom Wilson, Department of Geology and Geography Any general questions? These should look familiar to you and are a good place to start. Make sure you can do these.

We did these the other day in the “forward” direction Tom Wilson, Department of Geology and Geography

Note that both functions below have the same derivative Tom Wilson, Department of Geology and Geography Knowing the constant c is needed to correctly identify position along the y axis.

Were you able to work backwards to the parent functions? Tom Wilson, Department of Geology and Geography

In some cases you’ll get the starting function in a different format Tom Wilson, Department of Geology and Geography or or …

Both these functions have the same derivative but are part of the same family of curves Tom Wilson, Department of Geology and Geography

Take the simple function What do we have to differentiate to get x? What does the result represent? x y=x What is the area?

The integral in this case yields the area under the curve Tom Wilson, Department of Geology and Geography x y=x What does the constant c represent graphically and geometrically? An indefinite integral

Bring up Waltham’s file integ.xls Tom Wilson, Department of Geology and Geography Consider the integral of the function y=x. Compare areas estimated by summing a set of rectangles and that obtained by the actual integral.

Another comparison Tom Wilson, Department of Geology and Geography Approximation versus explicit integration

Add the areas of little rectangles Tom Wilson, Department of Geology and Geography Total Area under this Curve ≈

Discrete and analytic estimates of the integral/area Tom Wilson, Department of Geology and Geography In Waltham’s integ.xls file set n = 2 What is that “exact” number?

Tom Wilson, Department of Geology and Geography The exact number is obtained through integration with the integral derived using the power law for integrals

Briefly review the discrete development of the power rule for integration using an example function Tom Wilson, Department of Geology and Geography Then let or & so

Tom Wilson, Department of Geology and Geography These kinds of identities are not the sort of thing we geologists relish spending our time on. However, we can pull one out of the stack and put it to practical use and help develop a more intuitive understanding of the integral

Tom Wilson, Department of Geology and Geography The reason we do this is to get an idea whether there is some elegant way to describe this result as some other function substitute

Tom Wilson, Department of Geology and Geography Like most calculus concepts we are looking at behaviors when the interval becomes very small and the number of intervals becomes very large, so in the result below we just assume that n is much larger than 1 or 2 and simplify. From above with substitution yields or

Tom Wilson, Department of Geology and Geography What we find is that this area we are after can be expressed by a simple analytical function. So any time you want the area from 0 to some value X – you’ve got it! You are probably noticing something else interesting about this result as well.

This analysis suggests a general integration rule: the power rule for integrals Tom Wilson, Department of Geology and Geography You can see that the derivative This is an easy one. We just use the power law to find that = In general then:

Derivative in reverse = integral (area in this case) Tom Wilson, Department of Geology and Geography So all those reverse derivatives you’ve guessed are just areas under the curves representing those functions

Using the integral notation Tom Wilson, Department of Geology and Geography Like the derivative, this discrete summation can be carried into continuous form (that’s basically what we did when we assumed n>>1 or 2). The discrete sum is represented by the capital sigma and the continuous sum is expressed using the integral sign. The power rule for integrals is

Position as a function of velocity Tom Wilson, Department of Geology and Geography Let’s look at this problem using an example that may be easier to relate to. Let’s say that you know how fast something travels as a function of time and you want to catch it. For example, an asteroid is heading toward earth. You have to reach it before it gets to a certain distance from earth or it too late! The velocity keeps changing with time, but you know that it varies in proportion to time as:

Finding position when we know how v varies as a function of time Tom Wilson, Department of Geology and Geography Based on your calculations you determine that the asteroid (currently at position X o ) will reach the critical position (X) in time T. If you don’t get there at or before time T then you’ve got problems. How was X determined?

Since we know it’s velocity and acceleration, it is easy to compute x Tom Wilson, Department of Geology and Geography We know Integrating both sides yields You can check your result by taking the derivative What is the constant c?

Also note that the time it will take the asteroid, and thus your spacecraft, to get there will be Tom Wilson, Department of Geology and Geography No calculus required:

Tom Wilson, Department of Geology and Geography

The special case Tom Wilson, Department of Geology and Geography if then Thus When n = -1, what happens?

Tom Wilson, Department of Geology and Geography So this kind of integral is referred to as an indefinite integral. To overcome the problem we encountered with trying to find the moving target We need to have some additional information when we do the integration

Definite versus indefinite integrals Tom Wilson, Department of Geology and Geography

A structural geology problem cast in terms of calculus concepts 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

A structural geology problem cast in terms of calculus concepts Tom Wilson, Department of Geology and Geography Detachment horizon Given the analytic shapes of the deeper fold and shallower detached fold, how do we calculate the excess area in this cross sectional view?

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. Take a few moments to evaluate this integral and turn your result in.

Working through some of the additional group problems

Tom Wilson, Department of Geology and Geography Remember computer problems 8.13 and 8.14 are due today Finish reading chapter 9 and look over the problems in the text discussed in the applications section 9.6. How would you approach problems 9.7 and 9.8 Also look over questions 9.9 and They will be assigned … down the road Spend some more time working through those derivatives in reverse