Chapter 5: Applications of Integration
Section 5.7: Centroids
Centroid of a Plane Region
The centroid of a plane region R is its geometric center x&conjugate0;,y&conjugate0;. If a thin slab with uniform density and the shape of R were placed horizontal, it would balance on a support placed under the centroid.
Let Rx be a region R determined by fx≥gx,a≤x≤b, and let Ry be a region R determined by uy≥vy,α≤y≤β. Figures 5.7.1 and 5.7.2 are examples of regions Rx and Ry, respectively.
Figure 5.7.1 An example of a region Rx
Figure 5.7.2 An example of a region Ry
If R is a region Rx, then its centroid is defined as on the left in Table 5.7.1.
If R is a region Ry, then its centroid is defined as on the right in Table 5.7.1.
Centroid of Rx
Centroid of Ry
x&conjugate0;=1A∫abx fx−gx ⅆx
y&conjugate0;=1A∫αβy uy−vy ⅆy
Table 5.7.1 Formulas for computing the centroid of R
where A is the area of R.
Centroid of a Plane Curve
The centroid of a plane curve C is its geometric center x&conjugate0;,y&conjugate0;. If S is the arc length of the curve and ds the element of arc length, then the coordinates of the centroid are found via the formulas in Table 5.7.2.
Table 5.7.2 Coordinates of the centroid of a curve C
Determine the centroid of the region Rx bounded by fx=x and gx=x2.
By treating the region Rx in Example 5.7.1 as a region Ry, re-calculate the centroid.
Determine the centroid of the region Rx bounded by fx=cosx, gx=sinx, 0≤x≤π/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.
Determine the centroid of the trapezoid 25 × 15 × 15 × 15 (feet), longest edge uppermost and horizontal.
Determine the centroid of C, the upper half of the unit circle whose center is at the origin.
Determine the centroid of C, the parabola y=x2,x∈0,1.
Determine the centroid of C, the curve defined parametrically by the equations x=t cost, y=t2sint, t∈0,1.
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