©1999 BG Mobasseri18/25/2015 June 21, ‘99

©1999 BG Mobasseri28/25/2015 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

©1999 BG Mobasseri38/25/2015 HOW DOES MATLAB DO IT?  The primary function for evaluating definite integrals is quad8  quad8 has the following syntax – q=quad8(‘function’,a,b)  This is equivalent to the expression

©1999 BG Mobasseri48/25/2015 BUILT-IN MATLAB FUNCTIONS  Evaluate the following  Since cosine is a built-in MATLAB function; y=quad8(‘cos’,0,3*pi/2)

©1999 BG Mobasseri58/25/2015 USER-DEFINED FUNCTION (we haven’t 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,

©1999 BG Mobasseri68/25/2015 Try it! area under humps  humps is a pre-defined function in MATLAB. Let’s first plot it using x=0:0.01:2; plot(x,humps(x))  Now let’s find the area over a number of intervals – area=quad8(‘humps’,0,0.5) – area=quad8(‘humps’,0,1) – area=quad8(‘humps’,0,2)

©1999 BG Mobasseri78/25/2015 APPROXIMATION TO INTEGRALS - trapz  Function trapz uses areas of trapezoids to approximate the area under the curve x=0:0.01:2; y=humps(x) area=trapz(x,y) You might be surprised how large the answer is?answer next

©1999 BG Mobasseri88/25/2015 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? 1

©1999 BG Mobasseri98/25/2015 In all future slides... Use trapz in all future integration cases

©1999 BG Mobasseri108/25/2015 Try it !  Energy of a signal  Using trapz, find the energy of a gaussian pulse (slide 5) in the range (-1,1)

©1999 BG Mobasseri118/25/2015 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.

©1999 BG Mobasseri128/25/2015 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

©1999 BG Mobasseri138/25/2015 CROSS-SECTIONAL METHOD  Imagine sweeping a 1D shape, of varying cross sections A(x), along a path. This action will generate a swept volume. x b a c

©1999 BG Mobasseri148/25/2015 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 x h b

©1999 BG Mobasseri158/25/2015 THE DISC METHOD  Take a 1D curve f(x) and revolve it around the x- axis. This is a volume of revolution: – semi-circle---> sphere – triangle --> cone  Every cross section is a circle. The radius of the circle at x o is f(x o ).

©1999 BG Mobasseri168/25/2015 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

©1999 BG Mobasseri178/25/2015 THE SHELL METHOD  Define a function f(x) in a<x<b. Revolve R around the y-axis  Examples: – revolve a rectangle --> cylinder with a thickness – revolve a circle --> torus/donut y f(x) x R

©1999 BG Mobasseri188/25/2015 Try it!  Let f(x)=1-(x-2) 2 for 1<x<3. Revolve this around the y-axis and find its volume 31

©1999 BG Mobasseri198/25/2015 ARC LENGTH  Another important application of integrals is finding arc lengths x ab f(x)

©1999 BG Mobasseri208/25/2015 PARAMETRIC CURVES  It is frequently easier to work with a parametric representation of a curve,i.e. x=f(t) y=g(t)  For example, a circle x(t)=rcos(t) y(t)=rsin(t) r t

©1999 BG Mobasseri218/25/2015 LENGTH OF PARAMTERIC CURVES  Using derivatives of f(t) and g(t)

©1999 BG Mobasseri228/25/2015 CYCLOID P P P x full perimeter=2.pi.r  Path length traversed by a point on a wheel is of interest

©1999 BG Mobasseri238/25/2015 LENGTH OF A CYCLOID  The parametric equation of a cycloid with r=1 is given by x=t-sin(2.pi.t) y=1-cos(2.pi.t)  First, plot the cycloid for 0< t <1.  Then find its length for one cycle and compare it with the horizontal distance

©1999 BG Mobasseri248/25/2015 INTEGRALS IN POLAR COORDINATES  A curve can be represented in polar coordinates by  Equivalently   x y

©1999 BG Mobasseri258/25/2015 CURVE LENGTH  The length of a curve represented in polar coordinates is given by

©1999 BG Mobasseri268/25/2015 PERIMETER OF AN ELLIPSE  Find the perimeter of an ellipse given by

©1999 BG Mobasseri278/25/2015 cardioid, 3-leaved rose  cardioid is defined by r=1+cos .  3-leaved rose is given by r=cos3 .

©1999 BG Mobasseri288/25/2015 LENGTH OF A cardioid  We need the derivative of f(  – f’(  -sin(   Then,

©1999 BG Mobasseri298/25/2015 AREA IN POLAR COORDINATES f (  )

©1999 BG Mobasseri308/25/2015 AREA OF AN ELLIPSE  For the ellipse given by find its area and verify

©1999 BG Mobasseri318/25/2015 AREA OF A cardioid  Here we have  Then

©1999 BG Mobasseri328/25/2015 HOMEWORK-1  Find the length and area of a cardioid. Use the relevant equations for length and area given previously

©1999 BG Mobasseri338/25/2015 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.