Calculus

1
Calculus
Math Review with Matlab
Integration

• S. Awad, Ph.D.
• M. Corless, M.S.E.E.
• D. Cinpinski
• E.C.E. Department
• University of Michigan-Dearborn

2
Integration

• Definite and Indefinite Integrals
• Integral Pairs
• Closed Form Integrals
• Int Command
• Closed Form Example
• Multiple Independent Variable Example
• Numerical Integration
• Quad Command
• Numerical Integration Example
• Closed Form and Numerical Integration Example

3
Definite Indefinite Integrals

• A definite integral represents the area under a
  curve f(x) between the lower limit a and the
  upper limit b

• An integral is considered to be indefinite if the
  upper and lower limits are not specified

4
Integral Pairs

• Some indefinite integrals can be thought of as
  the inverse of differentiation

• A few common integral pairs are shown below

• Note that due to initial conditions, arbitrary
  integral pairs are not unique and may differ by a
  constant of integration, c

5
Closed Form Integrals

• Closed form integrals are integrals which can be
  expressed as explicit functions

• The integral pairs on the previous slide are
  examples of closed form integrals

• Techniques such as partial fraction expansion,
  integration by parts, and integration by
  substitution can be used to express some
  integrals in closed form

• It is not always possible to find the closed form
  for the integral of an arbitrary function

6
Evaluation of Definite Integral

• If a closed form indefinite integral exists, it
  can be used to evaluate a definite integral over
  a region of integration

• An integral from an arbitrary lower limit to a
  fixed upper limit is denoted as

• Thus definite integral from a lower limit a to an
  upper limit b can be evaluated using

7
Int Command

• The int command is used to solve integrals which
  can be expressed in closed form

• int(s) returns the indefinite integral of the
  symbolic variable s with respect to its default
  symbolic variable

int(s,v) returns the indefinite integral of the
symbolic variable s with respect to the symbolic
variable v
int(s,a,b) returns the definite integral of the
symbolic variable s with respect its default
symbolic variable from a to b
int(s,v,a,b) returns the definite integral of
the symbolic variable s with respect to the
symbolic variable v from a to b
8
Closed Form Example

• Given the function, f(x)

1) Use the int command to determine the Closed
Form Indefinite Integral, F(x)
2) From the closed form integral, F(x), determine
the definite integral of f(x) from 0 to p/2
3) Use the int command to directly determine the
definite integral of f(x) from 0 to p/2
4) Determine and plot a function A(z)
representing the general area under f from 0 to
any arbitrary point z
9
Indefinite Closed Form

• Matlab can be used to verify the integral pair
  shown on a previous slide
• The integration variable parameter does not need
  to be specified since x is the only variable in
  the expression, hence it will be the default
  variable of integration

syms x f(sin(2x))
Fint(f) F -1/2cos(2x)
10
Definite Integral Evaluation

• The definite integral can be evaluated from the
  indefinite integral using the relationship

Fbsubs(F,'x',pi/2) Fb 0.5000
Fasubs(F,'x',0) Fa -0.5000
A_pidiv2 Fb - Fa A_pidiv2 1
11
Matlab Direct Evaluation

• Since f(x) has a closed form, the int command can
  be used to directly determine the definite
  integral

f(sin(2x)) F_definiteint(f,0,pi/2)
F_definite 1
12
Area Under Curve

• A general function, A(z), used to plot the area
  under the curve f(x) to an arbitrary point z can
  be determined as follows

syms z F_z subs(F,x,z) F_0
subs(F,x,0) A_z F_z - F_0 A_z
-1/2cos(2z)1/2
A_z_pidiv2subs(A_z,z,pi/2) A_z_pidiv2 1
13
Graphical Definite Integral

• Area under f(x)
• from 0 to p/2 1

• Definite Integral Evaluation A(p/2)

subplot(2,1,1) ezplot(f,0,pi) grid on
subplot(2,1,2) ezplot(A_z,0,pi) grid on
14
Multiple Independent Variable Example

• Given the function f(t,x,y)

• Use Matlab to determine the closed form integrals
  with respect to the different independent
  variables

1) Integrate f with respect to t
2) Integrate f with respect to x
3) Integrate f with respect to y
15
Default Integration

• Integrating with respect to the default
  independent variable will integrate with respect
  to x

syms t x y fsym('t 2tx 3xyt') Fx
int(f)
Fx txtx23/2x2yt
16
Other Independent Variables

• Integration of f(t,x,y) with respect to t or y
  must be explicitly specified as an input argument
  to int

Ftint(f,t) Ft 1/2t2t2x3/2xyt2
Fyint(f,y) Fy ty2xyt3/2xy2t
17
Numerical Integration

• Some functions do not have closed form integrals

• A definite integral can be numerically
  approximated provided that the function is
  defined over the region of interest

• Numerical integration is performed by subdividing
  the integration region into very small regions,
  approximating the area in each region, and
  summing the areas

• If as the length of the subintervals tends to 0
  and the summation of areas tends to a unique
  limit, I, the definite integral over the interval
  is I

18
Quad Command

• The quad command is used to numerically evaluate
  an integral

Q quad('f',a,b) approximates the integral of
f(x) from a to b

• 'f' is a string containing the name of an
  m-function to be integrated. Function f must
  return a vector of output values when given an
  input vector.

• Q Inf is returned if an excessive recursion
  level is reached, indicating a possibly singular
  integral.

• Typically a new m-function will be created for f
  when numerically evaluating an integral

19
Numerical Integration Example

• Given the function, f(x)

• The definite integral, F

1) Use the symbolic toolbox to verify that the
f(x) does not have a closed form indefinite
integral
2) Plot f(x) to ensure that the function is
continuously defined over the integration region
3) Numerically integrate f(x) to determine F
20
No Closed Form

• No closed form can be found using Matlabs
  symbolic toolbox

f_sym sin(x)exp(-x2) int(f_sym) Warning
Explicit integral could not be found. gt In
C\MATLABR11\toolbox\symbolic\_at_sym\int.m at line
58 ans int(sin(x)exp(-x2),x)
21
Continuous Plot

• A plot verified that f(x) is continuous over the
  integration region

• Numerical integration is possible

ezplot(f_sym) grid on
22
Numerical Integration

• Must numerically evaluate the integral using the
  quad command which requires a function as an
  input
• Create a new m-function describing the function

• The function must return a vector so be sure to
  use . and . notations where appropriate

function f f_sin_exp_sqr( x ) fsin(x).exp(-x.
2)
23
Numerical Integration

• Use the quad command to perform the numerical
  integration and evaluate the definite integral

F_num_int quad('f_sin_exp_sqr',0,pi/2) F_num_
int 0.4024
24
Closed Form and Numerical Integration Example

• Given the function f(x)

• The definite integral, F

1) Plot f(x) to ensure that the function is
continuously defined over the integration region
2) Use the symbolic toolbox to evaluate the
definite integral
3) Numerically integrate f(x) to determine the
definite integral
25
Continuous Plot

• Use ezplot to plot the function
• Verified continuous from x0 to x4

syms x f_symexp(x)atan(x)
ezplot(f_sym) grid on
26
Symbolic Evaluation

• Use the symbolic toolbox int command to evaluate
  definite integral from the closed form integral

defint_symint(f_sym,0,4) F_sym
exp(4)4atan(4)-1/2log(17)-1
defint_dbldouble(F_sym) defint_dbl
57.4848
27
Numerical Evaluation

• Create a new m-function to represent f(x)

function f f_exp_atan( x ) fexp(x)atan(x)

• Use quad command to numerically evaluate the
  definite integral and verify the previous
  symbolic result

defint_numintquad('f_exp_atan',0,4) defint_num
int 57.4850

• Small difference to numerical approximation

28
Summary

• Definite integrals are evaluated over a
  continuous interval and result in a number

• Closed form indefinite integral functions exist
  for some functions independent of the integration
  limits

• The definite integrals can be evaluated from the
  closed form indefinite integral if it exists

• If a closed form indefinite integral does not
  exist, the definite integral of continuous
  functions can still be numerically approximated