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Geol 351 Geomath
Tom Wilson, Department of Geology and Geography
tom.h.wilsontom.wilson@mail.wvu.edu
Dept. Geology and GeographyWest Virginia University
Integral calculus - continued
Due dates
Tom Wilson, Department of Geology and Geography
• Hand in indefinite integrals problems (16).
• Finished reading chapter 9 ?
•Next week we’ll go after questions 9.9 and 9.10 look them over. Due dates forthcoming.
• The computer version of Question 9.7 will be assigned reviewed next time so look it over.
• Final exam will consist of in-class activities during the last week of class on April 25th and April 27th
Objectives for the day
Tom Wilson, Department of Geology and Geography
• Return homework/discuss
• Calculating lithostatic pressure (Sv) an integration
problem – problem introduction.
• 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 with in-class/take home and
assessment activities.
Questions 8.13-14
Tom Wilson, Department of Geology and Geography
Could you
actually present
the details of the
derivative??
A good self test
Tom Wilson, Department of Geology and Geography
1 2
1 2
max max
1 2
( ) ( )
t t
dS dS
dt dt
S Se e
How do you solve for t?
Can you show that …
Solve for t
1
2
1 2
ln
1 1t
Give it a try!
Alternative views of the fault problem
Tom Wilson, Department of Geology and Geography
sin cos
tew Lm
sin sin
t zw Lm
Two ways to minimize the “work” w=Lm
cos
em
Approach 1 Approach 2
t
We locate the zeros for the first derivative and know
they locate either a maximum or minimum
Tom Wilson, Department of Geology and Geography
At /4
Approach 1
Second derivatives
Tom Wilson, Department of Geology and Geography
2 4 4
2 3 3
sin cos
sin cos
d w
d
At 45o or /4, value
is positive so work
is a minimum
Approach 1
Where does the derivative =0
Tom Wilson, Department of Geology and Geography
At /2
2sin sin sin
t z t zw Lm
Approach 2
2nd derivative at /2 = 2 (positive) so a minimum
Tom Wilson, Department of Geology and Geography
The analysis gave us
two possible
solutions: one at 45o
and the other at 90o.
+ at /2=2
Approach 2
The vertical compressive stress (Sv) can be estimated using the
following integral which yields a result in units of F/A
Tom Wilson, Department of Geology and Geography
Sv=
t
e
m
L
( )* *b
a
forcedensity z g dz
area
oforce m g Vg =Mass x g (FORCE) of overburden/unit area
Equivalent to gh =force/area
Subsurface density varies continuously but is
sampled at discrete intervals in the density log
Tom Wilson, Department of Geology and Geography
P(z)
In continuous form, this is an integral of (z).
( )b
va
S z gdz
1
n
v i
i
S g z
In discrete form, this is a sum of I’s x …
Water column, ~1.03gm/cm3
Estimated density increase based on nearby logs
Log data 0.5ft sample rate
For now – let’s look at the in-class/take home
activity and assessment question
Tom Wilson, Department of Geology and Geography
Group discussion for a few minutes. Turn in next time.
For the definite integral
Tom Wilson, Department of Geology and Geography
2
2 2
2
2 2
bb
a a
xxdx
b a
y=x
x=a x=b
The area of the larger triangle
minus the smaller one.
Area of small
triangle =1/2 base
x height= a2/2.
Use class page link to Waltham Excel file
Integ.xlsx to experiment with these ideas
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
These are all definite integrals
Tom Wilson, Department of Geology and Geography
2 2 & y x x dx Approximation versus explicit integration
In these discrete approximations we are just
adding the areas of little rectangles
Tom Wilson, Department of Geology and Geography
x
( )f x
( )A f x x
Total Area under this Curve ≈ 1
( )n
i
f x x
Discrete and analytic estimates of
the integral/ the area
Tom Wilson, Department of Geology and Geography
In Waltham’s integ.xls file set n = 2
These examples provide illustrations of the ways computers can compute integrals. There’s usually some error, but we
can make that as small as we want by decreasing x.
Discrete integral computations allow you to solve
problems that don’t have simple integrals
Tom Wilson, Department of Geology and Geography
A simple integral with easily derived exact solution
11
1
n nx dx xn
the power rule for integrals
Tom Wilson, Department of Geology and Geography
3
3
d xc
dx
You can see that the derivative
3
3
d x
dx
This is an easy one. We just use the power law (in reverse: add 1 and divide by the new exponent) to find that
2x=
In general then the integral 11
1
n nx dx xn
The special case
Tom Wilson, Department of Geology and Geography
1 ln( )x dx x k
1
1
nxy
n
1
1
nn x
x dx kn
ndyx
dxIn general if then
Thus
When n = -1, you get 1/0!
See earlier lecture slides illustrating relationship between 1/x and ln(x).
1x dx
0
0
xc
As an indefinite integral
Tom Wilson, Department of Geology and Geography
1 ln( )x dx x k We add that constant to allow us to
accommodate arbitrary starting conditions.
2 2
11
1 ln( )x x
xxx dx x
As definite integral over defined limits
Definite versus indefinite integrals
Tom Wilson, Department of Geology and Geography
103 3 3
102
44
10 4312
3 3 3
xx dx
2y x
32
3
xx dx c
The indefinite
integral
Indefinite integrals provide general solutions
without specified range of integration
Tom Wilson, Department of Geology and Geography
32
3
xx dx C
2y x
Basic integration rules
Tom Wilson, Department of Geology and Geography
• power rule
• sum rule (distributive like the derivative)
• multiplication by a constant simply carries through as
with the derivative
• special case for
• integration of exponential functions
• integrals involving roots (use power rule and chain
rule)
• indefinite and definite evaluations of integrals
1x dx
3/22( )
3x adx x a C
Trig integrals in definite form
Tom Wilson, Department of Geology and Geography
32
2
cos( )a da
3sin( ) sin( )2 2
3
2
2
sin( )a
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
Rules overview
Tom Wilson, Department of Geology and Geography
( )af x dx
. . 3sin(x)dx=3 sin(x)dxe g
Given
where a is a constant; ( )a f x dx
a cannot be a function of x.
The constant factor rule for integration. If it’s not a function of x – pull it out.
Tom Wilson, Department of Geology and Geography
( ) ( ) ( )f x g x h x dx
( ) ( ) ( )f x dx g x dx h x dx
11
1
n nx dx x Cn
Just as with derivatives, distribute the integration through
the sum or difference of terms
The power rule has to be applied in reverse: add
one to the exponent and divide the function by
the exponent plus 1.
An integration problem to look over for next time
Tom Wilson, Department of Geology and Geography
Detachment
horizon
Detached rock forced
along the fault into a fold
We approximate the
shape of the deeper
fold as 24
425
s
xy
2
125
d
xy
We approximate the
shape of the
shallower detached
fold as
-x +x
Here we use two quadratics to represent deviations in relief of the
2nd order (shallow) fold relative to the 1st order (deep) regional fold
Tom Wilson, Department of Geology and Geography
Note that the limits used here coincide with the area for which relief on the upper blue
detached fold is greater than the orange fold in the lower unit (-5kft to 5kft)
244
25
x
2
125
x
The fold in the
overlying unit is a
fault propagation
fold. The Upper sheet
has additional
shortening associated
with this fold.
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?
244
25s
xy
2
125
d
xy
Calculate the area between these two curves
Tom Wilson, Department of Geology and Geography
Evaluate ( )
x
s dx
y y dx
2 25
5
44 1
25 25
x xdx
This is 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 minutes for group discussion.
Due end of next class.
Another geological application (see Section 9.6)
Tom Wilson, Department of Geology and Geography
Estimate the volume of material ejected during repeated eruptions of a volcano – in this case Mt. Fuji?
1
N
i
i
V V
2
1
N
i
i
r z
max
min
2Z
iZ
V r dz
Sum of flat circular disks
A sum of volume elements
Volume element z
Tom Wilson, Department of Geology and Geography
max
min
2Z
iZ
V r dz
2
ir dz
ridz
ri
is the volume of a disk having radius r and thickness dz.
=total volume
The sum of all disks with thickness dz
Area
Radius
2
1
N
i
i
V r z
go to let z dz
Tom Wilson, Department of Geology and Geography
2 2400 800400
3 3
z zr km
32
0
400 800400
3 3
z zV km
3 3 3
0 0 0
400 800400
3 3
z zV dz dz dz
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
32 1.5
0
400 800400
6 1.5 3
z zz
600 1600 1200
3200 628km
32
0iV r dz
The “definite” solution
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 …Radius of Mt. Fuji
0
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 15 20 25
Radius (km)
Ele
vati
on
(km
)
It works out pretty good though since the elevation at the foot of Mt.
Fuji is about 600-700 meters.
Lastly – take a look at these assessment
problems and hand in before leaving
Tom Wilson, Department of Geology and Geography
Looking ahead
Tom Wilson, Department of Geology and Geography
•Hand in indefinite integrals problems before leaving.
• Hand in the short assessment activity before leaving•We’ll take some time in the next class to go over the
lithostatic/hydrostatic pressure and fold area problems. They will be collected at the end of next class.
• Review questions 9.6 and 9.7 for next time•Have a look at problems 9.9 and 9.10
• 9.9 and 9.10 are tentatively due __TBA__.• The computer version of Question 9.7 will also be due on the
______to be announced______.
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