Lecture12Parametric Equations of Curves. Areas and ArcLengths for Parametric Curves12.1Parametric Equations of CurvesThere are several ways to represent curves analytically on theXOYplane.The first one is the explicitrepresentation we have met many times: in this case the curve is given in the formy=f(x),x∈[a,b]orx=g(y),y∈[c,d].But some curves cannot be given explicitly as functions ofxory. For example, the whole circle cannot begiven as a function ofxory1.Or, say, we want to calculate the length of the curve given in Figure12.1. Obviously, this curve cannot begiven explicitly2.-4-224-2-112Fig. 12.1: The parametric curvex=cost+5 sint,y=cos 5t+sint,t∈[0, 2π].We can handle this type of curves by specifying them in parametric form.Definition 12.1We will say that we are given a parametric curve, if we are given two continuous function f,g∈C[a,b]. We will denote that curve asx=f(t),y=g(t),t∈[a,b].Example:Here are some parametric curves:•x=cost,y=sint,t∈[0, 2π];•x=t2+3 sint,y=cost2-2tsin(et),t∈[0, 1];1Well, we can split it into upper and lower semi-circles and describe these two parts as functions ofx. Say, if our circle has a radius 2and center(0, 0), i.e. the circle is given byx2+y2=4,then the upper semicircle can be given byy=√4-x2,x∈[-2, 2]and the lower semi-circle can be given byy=-√4-x2,x∈[-2, 2].But then we need to deal with these two functions separately, if we want to do some calculations for the whole circle.2Again here we can split our curve into portions that can be described explicitly, but this is not the best way to deal with this kind ofproblems.58
Parametric Equations of Curves // M. Poghosyan, Calculus59•x=tarctgt,y=t+ln(5+sint),t∈R.To plot the parametric curvex=f(t),y=g(t),t∈[a,b],we take anyt∈[a,b], calculatex=f(t)andy=g(t)for thatt, and then plot on the coordinate plane thepoint(x,y) = (f(t),g(t)). Then we take anothert, do the same thing at that point and so on. In fact, we needto lettrun fromatob, and plot the obtained points(f(t),g(t))on the plane. Of course, there are infinitelymany real values oftbetweenaandb, so we need to plot infinitely many points actually, but in practice wetake a finite number of values oft, as dense as possible, and plot the corresponding points(f(t),g(t)), thenjoin this points smoothly to get the graph of our curve3. Please note that the parametertdoes not actuallymeet in the graph of parametric curve, since we plot onlyxandyvalues (obtained by evaluatingf(t)andg(t)att) on theXOYplane.Example:Here are the graphs of the above parametric curves, obtained byWol f ram Mathematicapackage:-1.0-0.50.51.0-1.0-0.50.51.0Fig. 12.2: The parametric curve (circle)x=cost,y=sint,t∈[0, 2π].