Chapter 6 | Applications of Integration
711
Figure 6.69 A representative rectangle of the region between two functions.
Note that the centroid of this rectangle is ⎛ ⎝ x i * , ⎛
⎞ ⎠ . We won’t go through all the details of the Riemann
⎝ f ( x i * )+ g ( x i * ) ⎞
⎠ /2
sum development, but let’s look at some of the key steps. In the development of the formulas for the mass of the lamina and the moment with respect to the y -axis, the height of each rectangle is given by f ( x i * )− g ( x i * ), which leads to the expression f ( x )− g ( x ) in the integrands. In the development of the formula for the moment with respect to the x -axis, the moment of each rectangle is found by multiplying the area of the rectangle, ρ ⎡ ⎣ f ( x i * )− g ( x i * ) ⎤ ⎦ Δ x , by the distance of the centroid from the x -axis, ⎛ ⎝ f ( x i * )+ g ( x i * ) ⎞ ⎠ /2, which gives ρ (1/2) ⎧ ⎩ ⎨ ⎡ ⎣ f ( x i * ) ⎤ ⎦ 2 − ⎡ ⎣ g ( x i * ) ⎤ ⎦ 2 ⎫ ⎭ ⎬ Δ x . Summarizing these findings, we arrive at the
following theorem.
Theorem 6.13: Center of Mass of a Lamina Bounded by Two Functions Let R denote a region bounded above by the graph of a continuous function f ( x ), below by the graph of the continuous function g ( x ), and on the left and right by the lines x = a and x = b , respectively. Let ρ denote the density of the associated lamina. Then we can make the following statements: i. The mass of the lamina is (6.21) m = ρ ∫ ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx . ii. The moments M x and M y of the lamina with respect to the x - and y -axes, respectively, are a b
(6.22)
b
⎞ ⎠ dx and M y = ρ ∫ a b x ⎡
⎛ ⎝ ⎡
M x = ρ ∫
1 2
⎤ ⎦ 2 −
⎤ ⎦ 2
⎣ f ( x )− g ( x ) ⎤
⎡ ⎣ g ( x )
⎣ f ( x )
⎦ dx .
a
⎝ x – , y –
iii. The coordinates of the center of mass ⎛
⎞ ⎠ are
M y m and y
(6.23)
– = M x m .
x – =
We illustrate this theorem in the following example.
Example 6.32 Finding the Centroid of a Region Bounded by Two Functions
Let R be the region bounded above by the graph of the function f ( x ) =1− x 2 and below by the graph of the function g ( x ) = x −1. Find the centroid of the region.
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