Chapter 5: Applications of Integration
Section 5.7: Centroids
Example 5.7.7
Determine the centroid of C, the parabola y=x2,x∈0,1.
Solution
Define the function y
Context Panel: Assign Function
yx=x2→assign as functiony
Calculate the arc length S
Expression palette: Definite Integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻S
∫011+y′x2ⅆx = 125−14ln−2+5→assign to a nameS
Calculate x&conjugate0;
Expression palette: Definite Integral template Write the formula for x&conjugate0;.
Context Panel: Approximate≻10 (digits)
1S∫01x1+y′x2 ⅆx = 5125−112125−14ln−2+5→at 10 digits0.5736270696
Calculate y&conjugate0;
Expression palette: Definite Integral template Write the formula for y&conjugate0;.
1S∫01yx1+y′x2 ⅆx = −132ln2+9325−164ln12+145125−14ln−2+5→at 10 digits0.4099802175
Figure 5.7.7(a) contains a graph of the parabola yx, and its centroid (red dot).
Note that again, the centroid of the curve does not lie on the curve. This was true for the upper half of the circle in Example 5.7.6, but there, y&conjugate0;=2/π<1 so it was pretty obvious that the centroid would not lie on the circle.
use plots in module() local p1,p2; p1 := plot(x^2, x = 0 .. 1, color = black); p2 := plot([[.5736270689, .4099802174]], style = point, symbol = solidcircle, symbolsize = 25, color = red); print(display(p1, p2, scaling = constrained, tickmarks = [3, 3],labels=[x,y])); end module: end use:
Figure 5.7.7(a) Parabola and its centroid
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