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Chapter 5: Applications of Integration
Section 5.7: Centroids
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Example 5.7.8
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Determine the centroid of , the curve defined parametrically by the equations , .
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Solution
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The expression for in the case of a parametrically defined curve is given by the last entry in Table 5.4.1. Unfortunately, for this curve , this expression is so unwieldy that the resulting integrals cannot be evaluated in closed form. Hence, all the ensuing integrals are evaluated numerically.
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Initialize
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Context Panel: Assign Function
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Context Panel: Assign Function
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Calculate the arc length
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Expression palette: Definite Integral template
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Context Panel: 2-D Math≻Convert To≻Inert Form
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Context Panel: Approximate≻10 (digits)
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Context Panel: Assign to a Name≻
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Calculate
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Expression palette: Definite Integral template
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Context Panel: 2-D Math≻Convert To≻Inert Form
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Context Panel: Approximate≻10 (digits)
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Calculate
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Expression palette: Definite Integral template
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Context Panel: 2-D Math≻Convert To≻Inert Form
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Context Panel: Approximate≻10 (digits)
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Figure 5.7.8(a) contains a graph of the parametrically defined curve , and its centroid (red dot).
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Note that again, the centroid of the curve does not lie on the curve.
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>
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use plots in
module()
local p1,p2,X,Y;
X:=t*cos(t);
Y:=t^2*sin(t);
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]));
end module:
end use:
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Figure 5.7.8(a) Curve and its centroid
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