Quiz #2 Arc Length --- Fall 2007
Contents
Exercise #1 --- Close all open figure windows.
The following command will close all open figure windows. This is something we do so that running the file will popup a fresh new figure window and we won't have to use the mouse or the command shg to bring the figure window to the front of the desktop.
close all
The domain.
We use the linspace command to produce 200 equally spaced points on the domain [1,4].
t=linspace(1,4,200);
Compute x and y.
We next compute x and y. Note the use of the array operators (dot-notation).
x=exp(t).*(cos(t)+sin(t)); y=exp(t).*(cos(t)-sin(t));
The plot.
We plot the result in the xy-plane
plot(x,y)
Annotate the axes
We use the xlabel and ylabel commands to annotate the axes.
xlabel('x-axis') ylabel('y-axis')
Title
The parametric equations are too long to fit in a one-line title. Hence, we will create a cell array that holds each equation as a row. The title command then takes the cell array and typesets it as a multi-line title.
Note the use of braces in titleStr(1) = { }.
There's a bit of TeX code here. The caret is used for superscripts in TeX. If you wish more than one character to be in the superscript, use braces to surround the intended superscript.
titleStr(1) = {'x = e^t ( cos(t) + sin(t) )'};
titleStr(2) = {'y = e^t ( cos(t) - sin(t) )'};
title(titleStr)
Smybolic Computation
In this segment, we will use the Symbolic Toolbox, Matlab's interface to Maple to make some symbolic calculations to check our hand-written evaluation of the integral.
First define t as a symbolic variable, then define x and y in terms of t.
syms t
x=exp(t)*(cos(t)+sin(t));
y=exp(t)*(cos(t)-sin(t));
Define ds
Define the fundamental piece of arc length and simplify the result.
ds=sqrt(diff(x,t)^2+diff(y,t)^2); ds=simple(ds)
ds = 2*exp(2*t)^(1/2)
Integrate
We now use the int command to integate
int(ds, 1, 4)
ans = 2*exp(4)-2*exp(1)
Click on the link that follows to obtain hand-calculations for the arc length.
Exercise #2 --- Close all open figure windows.
The following command will close all open figure windows. This is something we do so that running the file will popup a fresh new figure window and we won't have to use the mouse or the command shg to bring the figure window to the front of the desktop.
close all
The domain.
We use the linspace command to produce 200 equally spaced points on the domain [1,2].
t=linspace(1,2,200);
Compute x and y.
We next compute x and y. Note the use of the array operators (dot-notation).
x=t-t.^2; y=4/3*t.^(3/2);
The plot.
We plot the result in the xy-plane
plot(x,y)
Annotate the axes
We use the xlabel and ylabel commands to annotate the axes.
xlabel('x-axis') ylabel('y-axis')
Title
The parametric equations are too long to fit in a one-line title. Hence, we will create a cell array that holds each equation as a row. The title command then takes the cell array and typesets it as a multi-line title.
Note the use of braces in titleStr(1) = { }.
There's a bit of TeX code here. The caret is used for superscripts in TeX. If you wish more than one character to be in the superscript, use braces to surround the intended superscript.
titleStr(1) = {'x = t - t^2'};
titleStr(2) = {'y = (4/3) t^{3/2}'};
title(titleStr)
Numerical Integration
We will use an anonymous function to define the integrand
f = @(t) sqrt(1+4*t.^2);
Click on the link that follows to obtain hand-calculations for the integrand..
Quad
We now use the quad command to integrate
quad(f,1,2)
ans =
3.1678