Length of a curve:
The length of curve C with equation as the limit of the length of these inscribed polygons (if the limit exists)

The arc length formula:
If is continuous on then the length of the curve is

Definition of the surface area:
If is a positive function which has a continuous derivative then the surface area obtained by rotating the curve about the x-axis is given by the following:

Definition of the force F of an object:
If an object moves along a straight line with position function s(t), the force F on the object given by Newton’s Second Law of Motion is:

Definition of the center of mass:
For a system of particles, with massand the location of on the axis, the center of mass of the system is at:  where is the total mass of the system.

Pappus’s Theorem for Volumes:
If a plane region is revolved once about a line in the plane that does not cut through the region’s interior.

This tutorial introduces the concepts of arc length, center of mass and fluids. The examples given in this tutorial allow for the development of the skills needed to find the arc length of a smooth curve, to find the area of a surface of a revolution and to find the work done by a constant or variable force.

Finding the area of a surface of a revolution is shown with the use of the concept of cutting a solid into separate pieces. Surface area formulas are summarized with the help of the arc length concept. The work done by a constant or variable force is presented with the use of integrals and graphical examples. Applications of the work done by an object are shown with the use of Hooke’s Law.

Specific Tutorial Features:

• Step by step examples of the different aspects of arc length, work, center of mass and fluids
  
• Visual examples are given to emphasize the center of mass ideals and the theorem of pappus to find the volume of a solid revolution.

Series Features:

• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
  
• Definition slides introduce terms as they are needed.
  
• Visual representation of concepts
  
• Animated examples—worked out step by step
  
• A concise summary is given at the conclusion of the tutorial.

Finding the length of a smooth curve
      Definition of the length of a curve
      The arc length formula
Finding the area of a surface of a revolution
      Definition of surface area
Finding the work done by a constant or variable force
      Definition of force
      Application of Hooke’s Law
Finding the center of mass
Use the Theorem of pappus to find the volume of a solid revolution
Finding fluid pressure and fluid force