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Subsections

More Integration Topics

Background

For Exercise 1 .....

Except for specially constructed textbook exercises, an arclength problem integrand rarely has an elementary antiderivative and so the integral for the problem must be evaluated through approximation methods. As we have seen, the Maple evalf command invokes a powerful approximation routine. However, if you use int followed by evalf, Maple will first try to find an antiderivative. Since arclength integrals can be very nasty Maple may spend a long time trying to execute int. A better approach will usually be to use Int followed by evalf. This will tell Maple to move directly to an approximation and, when the output from Int is displayed, you will have a chance to check that you have correctly typed the integrand.

For Exercise 2 .....

We expand on what is given in Section 6.5 of the text and discuss work along a curve. Considering what a Riemann sum would look like when the basic $W = F \cdot d$ is used, it should be easy to see that the formula for the work done along a curve is
\begin{maplelatex} \begin{displaymath} \mbox{Work} = \displaystyle\int^b_a F(x) \sqrt{1 + (h^\prime(x))^2}\;dx,\end{displaymath}\end{maplelatex}
where y = h(x) defines the curve on the interval [a,b],

For Exercise 3 .....

Parametric equations are introduced in Section 6.4 of the text and further discussed in Section 13.1. Parametric representations of functions are often very useful in problems describing the motion of an object.

The parametric presentation given as
\begin{maplelatex} \begin{displaymath} x = f(t),\;\;y = g(t),\;\;a \leq t \leq b\end{displaymath}\end{maplelatex}
can be plotted in this way. 

  > plot([f(t),g(t),t=a..b]);
The square brackets alert Maple to the fact that this is a parametric presentation. Sometimes these graphs are very complicated and it is desirable to be able to see the graph traced out point by point as t increases. This can be done by ParamPlot, a command that incorporates the Maple animate command. Note that you must first load the CalcP package. 
  > with(CalcP);

  > ParamPlot([f(t),g(t)],t = a..b);

Here is a description of a rather famous curve. When a circle of radius b rolls (without slipping) along the inside of another circle of radius a > b, the curve traced out by a fixed point P on the smaller circle is called a hypocycloid. If the initial position of point P is (a,0), then the hypocycloid has parametric equations
\begin{maplelatex} \begin{displaymath} x(t) = (a - b)\cos(t) + b\cos\left(\displaystyle\frac{a - b}{b}\;t\right),\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} y(t) = (a - b)\sin(t) - b\sin\left(\displaystyle\frac{a - b}{b}\;t\right).\end{displaymath}\end{maplelatex}

To investigate multiple examples of hypocycloids you could try the following 

  > x:=t->(a-b)*cos(t)+b*cos((a-b)*t/b);

  > y:=t->(a-b)*sin(t)-b*sin((a-b)*t/b);

  > a:=1;b:=4;

  > plot([x(t),y(t),t=0..2*Pi]);
Then, to do another case, declare the new values of a and b and give the plot command with the correct t range. 
  > a:=7;b:=2;

  > plot([x(t),y(t),t=0..4*Pi]);

For Exercise 4 .....

Here is another well known use of parametric equations. The trajectory of a projectile fired from the origin with initial speed s0 and angle of elevation $\theta$ is described parametrically by
\begin{maplelatex} \begin{displaymath} x(t) = s_0(\cos(\theta))t,\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} y(t) = -\displaystyle\frac{1}{2} gt^2 + s_0(\sin(\theta))t,\end{displaymath}\end{maplelatex}
where g is a gravity constant.

For Exercise 5 .....

The techniques needed for this problem were discussed in class.

Exercises

1.
(a)
A cable for a suspension bridge has the shape of a parabola. Assume the equation of the parabola is y = kx2. The suspension bridge has total span 2S and the height of the cable (relative to its lowest point) is H at each end. Develop the integral L that gives a formula for the length of the cable in terms of S and H. (You may do this part by hand on a page attached to the lab report.)
(b)
Italian engineers have proposed a single-span suspension bridge across the Strait of Messina (8 km wide) between Italy and Sicily. The plans include suspension towers 380m high at each end. Use the formula for L developed in (a) to find the length of the parabolic suspension cable for this bridge. Do you have any comments on the design?
2.
Find the work done along the curve $h(x) = \sin(3x)$ on the interval $[0,\pi]$ if F(x) = 2x2. (If your evalf command runs more than 15 seconds or so, you should probably stop the execution and reconsider the syntax you are using for the derivative.)

3.
Investigate how the values of a and b influence the shape of the graph of a hypocycloid by plotting the following cases.
\begin{maplelatex} \begin{displaymath} a = 4,\;\;b = 1,\;\;0 \leq t \leq 2\pi\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} a = 7,\;\;b = 2,\;\;0 \leq t \leq 4\pi\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} a = 9,\;\;b = 2.\;\; 0 \leq t \leq 4\pi\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} a = 11,\;\;b = 3,\;\;0 \leq t \leq 6\pi\end{displaymath}\end{maplelatex}

\begin{maplelatex} \begin{displaymath} a = 33,\;\;b = 13,\;\;0 \leq t \leq 26\pi\end{displaymath}\end{maplelatex}
Write a paragraph describing what you see and telling what conclusions can be drawn.

4.
A projectile is fired (from the origin) with an initial speed of 160 ft/sec and a $30^\circ$ angle of elevation. Find the length of the path of flight of the projectile. Assume that the terrain is absolutely flat. Use 32 ft/sec2 for the value of g.

5.
Consider the function given parametrically by
\begin{maplelatex} \begin{displaymath} x = t - 2\sin(t),\;\;y = 2 - 2\cos(t).\end{displaymath}\end{maplelatex}
(a)
Find the area under the curve for t in [2,5].
(b)
Find the volume of the solid generated by revolving the area of part (a) about the x-axis.
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Next: About this document ... Up: Labs and Projects for Previous: Labs and Projects for

Christine Marie Bonini
4/15/1999