Chapter 6 | Applications of Integration
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6.5 | Physical Applications
Learning Objectives 6.5.1 Determine the mass of a one-dimensional object from its linear density function. 6.5.2 Determine the mass of a two-dimensional circular object from its radial density function. 6.5.3 Calculate the work done by a variable force acting along a line. 6.5.4 Calculate the work done in pumping a liquid from one height to another. 6.5.5 Find the hydrostatic force against a submerged vertical plate. In this section, we examine some physical applications of integration. Let’s begin with a look at calculating mass from a density function. We then turn our attention to work, and close the section with a study of hydrostatic force. Mass and Density We can use integration to develop a formula for calculating mass based on a density function. First we consider a thin rod or wire. Orient the rod so it aligns with the x -axis, with the left end of the rod at x = a and the right end of the rod at x = b ( Figure 6.48 ). Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.
Figure 6.48 We can calculate the mass of a thin rod oriented along the x -axis by integrating its density function.
If the rod has constant density ρ , given in terms of mass per unit length, then the mass of the rod is just the product of the density and the length of the rod: ( b − a ) ρ . If the density of the rod is not constant, however, the problem becomes a little more challenging. When the density of the rod varies from point to point, we use a linear density function , ρ ( x ), todenote the density of the rod at any point, x . Let ρ ( x ) be an integrable linear density function. Now, 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 ]. Figure 6.49 shows a representative segment of the rod.
Figure 6.49 A representative segment of the rod.
The mass m i of the segment of the rod from x i −1 to x i is approximated by m i ≈ ρ ( x i * )( x i − x i −1 ) = ρ ( x i * )Δ x . Adding the masses of all the segments gives us an approximation for the mass of the entire rod:
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