Chapter 5: Applications of Integration
Section 5.7: Centroids
Determine the centroid of C, the curve defined parametrically by the equations x=t cost,y=t2sint, t∈0,1.
The expression for ds in the case of a parametrically defined curve is given by the last entry in Table 5.4.1. Unfortunately, for this curve C, this expression is so unwieldy that the resulting integrals cannot be evaluated in closed form. Hence, all the ensuing integrals are evaluated numerically.
Context Panel: Assign Function
xt=t cost→assign as functionx
yt=t2sint→assign as functiony
Calculate the arc length S
Expression palette: Definite Integral template
Context Panel: 2-D Math≻Convert To≻Inert Form
Context Panel: Approximate≻10 (digits)
Context Panel: Assign to a Name≻S
∫01x′t2+y′t2ⅆt→at 10 digits1.218980671→assign to a nameS
1S∫01xtx′t2+y′t2 ⅆt→at 10 digits0.4182510455
1S∫01ytx′t2+y′t2 ⅆt→at 10 digits0.3007887663
Figure 5.7.8(a) contains a graph of the parametrically defined curve C, and its centroid (red dot).
Note that again, the centroid of the curve does not lie on the curve.
use plots in
p1 := plot([X,Y,t=0..1], color = black);
p2 := plot([[0.4182510455, 0.3007887662]], style = point, symbol = solidcircle, symbolsize = 25, color = red);
print(display(p1, p2, scaling = constrained,labels=[x,y],tickmarks=[3,4]));
Figure 5.7.8(a) Curve C and its centroid
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