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Chapter 6 | Applications of Integration
Example 6.24 Calculating Mass from Radial Density
Let ρ ( x ) = x represent the radial density of a disk. Calculate the mass of a disk of radius 4.
Solution Applying the formula, we find
r 2 πxρ ( x ) dx
m = ∫
0
4 2 πx xdx =2 π ∫ 0 4
= ∫
x 3/2 dx
0
x 5/2 |
0 4
= 4 π 5
[32] = 128 π 5 .
=2 π 2 5
Let ρ ( x ) =3 x +2 represent the radial density of a disk. Calculate the mass of a disk of radius 2.
6.24
Work Done by a Force We now consider work. In physics, work is related to force, which is often intuitively defined as a push or pull on an object. When a force moves an object, we say the force does work on the object. In other words, work can be thought of as the amount of energy it takes to move an object. According to physics, when we have a constant force, work can be expressed as the product of force and distance. In the English system, the unit of force is the pound and the unit of distance is the foot, so work is given in foot-pounds. In the metric system, kilograms and meters are used. One newton is the force needed to accelerate 1 kilogram of mass at the rate of 1 m/sec 2 . Thus, the most common unit of work is the newton-meter. This same unit is also called the joule . Both are defined as kilograms times meters squared over seconds squared ⎛ ⎝ kg·m 2 /s 2 ⎞ ⎠ . When we have a constant force, things are pretty easy. It is rare, however, for a force to be constant. The work done to compress (or elongate) a spring, for example, varies depending on how far the spring has already been compressed (or stretched). We look at springs in more detail later in this section. Suppose we have a variable force F ( x ) that moves an object in a positive direction along the x -axis from point a to point b . To calculate the work done, we partition the interval ⎡ ⎣ a , b ⎤ ⎦ and estimate the work done over each subinterval. So, for i =0, 1, 2,…, n , let P ={ x i } be a regular partition of the interval ⎡ ⎣ a , b ⎤ ⎦ , and for i =1, 2,…, n , choose an arbitrary point x i * ∈ [ x i −1 , x i ]. To calculate the work done to move an object from point x i −1 to point x i , we assume the force is roughly constant over the interval, and use F ( x i * ) to approximate the force. The work done over the interval [ x i −1 , x i ], then, is given by W i ≈ F ( x i * )( x i − x i −1 ) = F ( x i * )Δ x . Therefore, the work done over the interval ⎡ ⎣ a , b ⎤ ⎦ is approximately W = ∑ i =1 n W i ≈ ∑ i =1 n F ( x i * )Δ x . Taking the limit of this expression as n →∞ gives us the exact value for work:
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