Chapter 5: Applications of Integration
Section 5.7: Centroids
Example 5.7.4
Find the centroid of T, the triangle whose vertices are 0,0,0,a,b,c. Note that T forms a region Rx.
Show that the centroid is the intersection of the medians.
Solution
Mathematical Solution
Part (a)
The area of T is A=ab/2 (half base times height).
The bounding edges of T are
fx=c−ab x+a and gx=cb x
The centroid is given by
x&conjugate0;=1A∫0bx fx−gx ⅆx=b/3
y&conjugate0;=1A∫0bf2x−g2x/2 ⅆx=a+c/3
Figure 5.7.4(a) Centroid and medians of T
The centroid is therefore the point b/3,a+c/3.
Part (b)
The medians yk,k=1,2,3, to the midpoints mk,k=1,2,3, in Figure 5.7.4(a) are respectively
y1=a+cb x
y2=c−2 ab x+a
y3=2 c−a2 b x+a2
The intersection of y1 and y2 is b/3,a+c/3, the centroid. Moreover, evaluating y3 at x=b/3 gives a+c/3, so all three medians meet at the centroid.
Maple Solution
Initialize
Tools≻Load Package: Student Precalculus
Loading Student:-Precalculus
Obtain the area of T
Context Panel: Assign to a Name≻A
a b/2→assign to a nameA
Obtain fx
Apply the Line command from the Student Precalculus package.
Line0,a,b,c1
y=−a−cxb+a
Write fx=… Context Panel: Assign Function
fx=−a−cxb+a→assign as functionf
Obtain gx
Line0,0,b,c1
y=xcb
Write gx=… Context Panel: Assign Function
gx=cb x→assign as functiong
Calculate x&conjugate0;
Expression palette: Definite-integral template Press the Enter key.
Context Panel: Simplify≻Simplify
1A∫0bx fx−gx ⅆx
213−a−cb−cbb3+12ab2ab
= simplify
13b
Calculate y&conjugate0;
1A∫0bf2x−g2x/2 ⅆx
21312a−c2b2−12c2b2b3−12aa−cb+12a2bab
13a+13c
Thus, the centroid is the point b/3,a+c/3.
Next, obtain the midpoints of the three sides of T and the three medians of T, then intersect the medians.
Midpoints
Apply the Midpoint command from the Student Precalculus package.
m1≔Midpoint0,a,b,c = 12b,12a+12c
m2≔Midpoint0,0,b,c = 12b,12c
m3≔Midpoint0,0,0,a = 0,12a
Medians
M1≔Line0,0,m11
y=−2−12c−12axb
M2≔Line0,a,m21
y=−2a−12cxb+a
M3≔Lineb,c,m31
y=c−12axb+12a
Intersect medians M1 and M2
Apply the solve command.
solveM1,M2,x,y
x=13b,y=13a+13c
Evaluate median M3 at x=b/3
Write M3 and press the Enter key.
Context Panel: Evaluate at a Point≻x=b/3
M3
→evaluate at point
y=13a+13c
Medians M1 and M2 intersect at the centroid, and median M3 also passes through the centroid.
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