June 21, 99

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June 21, 99
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EVALUATING DEFINITE INTEGRALS

• MATLAB can evaluate definite integrals like
• This is provided that the integrand f(x) be
  available as a function, not an array of numbers

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HOW DOES MATLAB DO IT?

• The primary function for evaluating definite
  integrals is quad8
• quad8 has the following syntax
• qquad8(function,a,b)
• This is equivalent to the expression

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BUILT-IN MATLAB FUNCTIONS

• Evaluate the following
• Since cosine is a built-in MATLAB function
• yquad8(cos,0,3pi/2)

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USER-DEFINED FUNCTION(we havent covered
function writing)

• You can integrate functions that are not part of
  MATLAB library.
• For example, you can write a function of your own
  such as gauss,

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Try it! area under humps

• humps is a pre-defined function in MATLAB. Lets
  first plot it using
• x00.012
• plot(x,humps(x))
• Now lets find the area over a number of
  intervals
• areaquad8(humps,0,0.5)
• areaquad8(humps,0,1)
• areaquad8(humps,0,2)

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APPROXIMATION TO INTEGRALS - trapz

• Function trapz uses areas of trapezoids to
  approximate the area under the curve

x00.012 yhumps(x) areatrapz(x,y)
You might be surprised how large the
answer is?answer next
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Sample Spacing

• trapz assumes unit spacing between samples
• If that is not true, the output of trapz must be
  scaled by the actual spacing, e.g. 0.1
• So what is the right answer in the previous slide?

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In all future slides...
Use trapz in all future integration cases
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Try it !

• Energy of a signal
• Using trapz, find the energy of a gaussian pulse
  (slide 5) in the range (-1,1)

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EXTENSION OF 1D INTEGRALS

• 1-D integral can geometrically be interpreted as
  an area.
• It is possible to evaluate volumes, not by
  multidimensional integrals as is generally done ,
  but as 1-D integrals.

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DEFINING VOLUMES

• There are a number of ways a 3D shape can be
  generated
• Sweeping a Cross Section
• The Disc Method
• The Washer Method
• The Shell Method

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CROSS-SECTIONAL METHOD

• Imagine sweeping a 1D shape, of varying cross
  sections A(x), along a path. This action will
  generate a swept volume.

c
b
a
x
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VOLUME OF A PYRAMID

• In problems like this you must first do two
  things
• write a function for the cross section as a
  function of x
• determine the lower and upper limit of the sweep

h
x
b
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THE DISC METHOD

• Take a 1D curve f(x) and revolve it around the
  x-axis. This is a volume of revolution
• semi-circle---gt sphere
• triangle --gt cone
• Every cross section is a circle. The radius of
  the circle at xo is f(xo).

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Try it! REVOLVING A SINUSOID

• Take one period of and
  revolve it around the x-axis. Plot the shape then
  find the volume of the revolution
  

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THE SHELL METHOD

• Define a function f(x) in altxltb. Revolve R
  around the y-axis
• Examples
• revolve a rectangle --gt cylinder with a thickness
• revolve a circle --gt torus/donut

y
f(x)
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Try it!

• Let f(x)1-(x-2)2 for 1ltxlt3. Revolve this around
  the y-axis and find its volume

3
1
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ARC LENGTH

• Another important application of integrals is
  finding arc lengths

f(x)
x
a
b
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PARAMETRIC CURVES

• It is frequently easier to work with a parametric
  representation of a curve,i.e.
• xf(t)
• yg(t)
• For example, a circle
• x(t)rcos(t)
• y(t)rsin(t)

r
t
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LENGTH OF PARAMTERIC CURVES

• Using derivatives of f(t) and g(t)

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CYCLOID

• Path length traversed by a point on a wheel is of
  interest

P
P
x
P
full perimeter2.pi.r
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LENGTH OF A CYCLOID

• The parametric equation of a cycloid with r1 is
  given by
• xt-sin(2.pi.t)
• y1-cos(2.pi.t)
• First, plot the cycloid for 0lt t lt1.
• Then find its length for one cycle and compare it
  with the horizontal distance

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INTEGRALS IN POLAR COORDINATES

• A curve can be represented in polar coordinates
  by
• Equivalently

???
y
???
x
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CURVE LENGTH

• The length of a curve represented in polar
  coordinates is given by

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PERIMETER OF AN ELLIPSE

• Find the perimeter of an ellipse given by

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cardioid, 3-leaved rose

• cardioid is defined by r1cos?.
• 3-leaved rose is given by rcos3?.

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LENGTH OF A cardioid

• We need the derivative of f(??
• f(????-sin(??
• Then,

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AREA IN POLAR COORDINATES
f (?)
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AREA OF AN ELLIPSE

• For the ellipse given by
• find its area and verify

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AREA OF A cardioid

• Here we have
• Then

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HOMEWORK-1

• Find the length and area of a cardioid. Use the
  relevant equations for length and area given
  previously

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HOMEWORK-2

• Find the energy of the bond clip using trapz.
  This routine assumes unit sample spacing.
  However, bond is sampled at 8KHz. Take this into
  account.