Lecture12 - Lecture 12 Parametric Equations of Curves Areas...

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L ecture 12 Parametric Equations of Curves. Areas and Arc Lengths for Parametric Curves 12.1 Parametric Equations of Curves There are several ways to represent curves analytically on the XOY plane. The first one is the explicit representation we have met many times: in this case the curve is given in the form y = f ( x ) , x [ a , b ] or x = g ( y ) , y [ c , d ] . But some curves cannot be given explicitly as functions of x or y . For example, the whole circle cannot be given as a function of x or y 1 . Or, say, we want to calculate the length of the curve given in Figure 12.1 . Obviously, this curve cannot be given explicitly 2 . - 4 - 2 2 4 - 2 - 1 1 2 Fig. 12.1 : The parametric curve x = cos t + 5 sin t , y = cos 5 t + sin t , t [ 0, 2 π ] . We can handle this type of curves by specifying them in parametric form. Definition 12.1 We 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 as x = f ( t ) , y = g ( t ) , t [ a , b ] . Example: Here are some parametric curves: x = cos t , y = sin t , t [ 0, 2 π ] ; x = t 2 + 3 sin t , y = cos t 2 - 2 t sin ( e t ) , t [ 0, 1 ] ; 1 Well, we can split it into upper and lower semi-circles and describe these two parts as functions of x . Say, if our circle has a radius 2 and center ( 0, 0 ) , i.e. the circle is given by x 2 + y 2 = 4, then the upper semicircle can be given by y = 4 - x 2 , x [ - 2, 2 ] and the lower semi-circle can be given by y = - 4 - x 2 , 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. 2 Again 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 of problems. 58
Parametric Equations of Curves // M. Poghosyan, Calculus 59 x = t arctg t , y = t + ln ( 5 + sin t ) , t R . To plot the parametric curve x = f ( t ) , y = g ( t ) , t [ a , b ] , we take any t [ a , b ] , calculate x = f ( t ) and y = g ( t ) for that t , and then plot on the coordinate plane the point ( x , y ) = ( f ( t ) , g ( t )) . Then we take another t , do the same thing at that point and so on. In fact, we need to let t run from a to b , and plot the obtained points ( f ( t ) , g ( t )) on the plane. Of course, there are infinitely many real values of t between a and b , so we need to plot infinitely many points actually, but in practice we take a finite number of values of t , as dense as possible, and plot the corresponding points ( f ( t ) , g ( t )) , then join this points smoothly to get the graph of our curve 3 . Please note that the parameter t does not actually meet in the graph of parametric curve, since we plot only x and y values (obtained by evaluating f ( t ) and g ( t ) at t ) on the XOY plane. Example: Here are the graphs of the above parametric curves, obtained by Wol f ram Mathematica package: - 1.0 - 0.5 0.5 1.0 - 1.0 - 0.5 0.5 1.0 Fig. 12.2 : The parametric curve (circle) x = cos t , y = sin t , t [ 0, 2 π ] .

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