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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 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.
> 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
To investigate multiple examples of hypocycloids you could try the following
> 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.
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 is described parametrically by
where g is a gravity constant.
The techniques needed for this problem were discussed in class.
Christine Marie Bonini