Chapter 7: Further Applications of Integration
In Chapter 5 we explored a few applications of integration, ranging from volumes to work. Of course, many other applications exist. In this chapter we incorporate the techniques of integration learned from Chapter 6 to solve more problems. We will first discuss a curve's arc length (7.1) and surface area of revolution (7.2). Then we will measure how much benefit consumers and producers each experience in a market in Section 7.3. Because physics is central to calculus, we will explore how to locate the point at which a plate balances (7.4) and the force acting on a dam (7.5). Finally, we will end this chapter by learning about what role calculus plays in probability and statistics (7.6).
Sections
7.1 Arc Length
Lengths of smooth, planar curves by integrating with \(x\) and \(y.\) Arc length functions. Derivation of formulas and geometric intuition with animation.7.2 Surface Areas of Revolution
The surface area of the solid generated when a bounded region is spun around an axis. Integration with both \(x\) and \(y\) and computing the general differential \(\dd s.\)7.3 Consumer Surplus and Producer Surplus
Introduction to microeconomics: discussion of demand and supply curves, and producer surplus and consumer surplus by integration.7.4 Moments and Centers of Mass
Definition of a moment and center of mass. Calculating moments and centers of mass for particle system. Integrating to find moments and centroids of laminae bounded by one or two curves. Theorem of Pappus.7.5 Hydrostatics
The hydrostatic force acting on a stationary, submerged body.7.6 Probability
Definition of probability density functions. Use of calculus to determine whether a function is a probability density function. Mean of a probability distribution. Exponential decay probability distributions and their means. The Normal distribution, its mean \(\mu,\) and its standard deviation \(\sigma.\)