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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.

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 is used, it should be easy to see that the
formula for the work done along a curve is

where *y* = *h*(*x*) defines the curve on the interval [*a*,*b*],

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

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

> 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

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:=7;b:=2;

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

Here is another well known use of parametric equations. The
trajectory of a projectile fired from the origin with initial speed
*s _{0}* and angle of elevation is described parametrically by

where

The techniques needed for this problem were discussed in class.

- 1.
- (a)
- A cable for a suspension bridge has the shape of a
parabola. Assume the equation of the parabola is
*y*=*kx*. The suspension bridge has total span 2S and the height of the cable (relative to its lowest point) is^{2}*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 on
the interval if
*F*(*x*) = 2*x*. (If your^{2}`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.

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 angle of elevation. Find the length of
the path of flight of the projectile. Assume that the terrain is
absolutely flat. Use 32 ft/sec
^{2}for the value of*g*. - 5.
- Consider the function given parametrically by

- (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.

4/15/1999