# 7: Applications of Integration

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- 7.1: Area Between Curves
- This chapter employs the following technique to a variety of applications. Suppose the value QQ of a quantity is to be calculated. We first approximate the value of QQ using a Riemann Sum, then find the exact value via a definite integral. This idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves.

- 7.2: Volume by Cross-Sectional Area: Disk and Washer Methods
- Given an arbitrary solid, we can approximate its volume by cutting it into nn thin slices. When the slices are thin, each slice can be approximated well by a general right cylinder. Thus the volume of each slice is approximately its cross-sectional area ×× thickness. (These slices are the differential elements.)

- 7.3: The Shell Method
- The previous section introduced the Disk and Washer Methods, which computed the volume of solids of revolution by integrating the cross--sectional area of the solid. This section develops another method of computing volume, the Shell Method. Instead of slicing the solid perpendicular to the axis of rotation creating cross-sections, we now slice it parallel to the axis of rotation, creating "shells."

- 7.4: Arc Length and Surface Area
- In this section, we address a simple question: Given a curve, what is its length? This is often referred to as arc length.

- 7.5: Work
- Work is the scientific term used to describe the action of a force which moves an object. The SI unit of force is the Newton (N), and the SI unit of distance is a meter (m). The fundamental unit of work is one Newton--meter, or a joule (J). That is, applying a force of one Newton for one meter performs one joule of work.

- 7.6: Fluid Forces
- In the unfortunate situation of a car driving into a body of water, the conventional wisdom is that the water pressure on the doors will quickly be so great that they will be effectively unopenable. How can this be true? How much force does it take to open the door of a submerged car? In this section we will find the answer to this question by examining the forces exerted by fluids.

### Contributors

Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/