•   Area of a Plane Region 
  Area played a fundamental role in the introduction of Riemann Sums and Definite Integrals.  The Fundamental Theorems of Calculus and techniques for Evaluating Definite Integrals provide the tools necessary for the efficient computation of the area of a region bounded by the graph of one or more functions.
•   Mean Value Theorem for Integrals 
  The Mean Value Theorem for Integrals is an integrated version of the Mean Value Theorem (for Derivatives).  Instead of talking about slopes of secant and tangent lines, the MVT for Integrals is stated in terms of areas and average values of a function.
•   Volume of a Solid 
  The ideas used to express the Area of a Plane Region as a Definite Integral can be extended to the Volume of a Solid.  The key is to be able to identify the area of every cross-sectional slice.
•   Length of a Plane Curve 
  Definite Integrals can be used to compute the Length of a Plane Curve by recognizing that the length can be written as the limit of a Riemann Sum.
•   Work 
  Work is the final example of a physical quantity that can be expressed as a Definite Integral.  As with all other applications in this unit, the key is to approximate the quantity with a Riemann Sum and to recognize the exact quantity as the limit of a Riemann Sum, i.e., as a .htmlref="http://www.math.sc.edu/~meade/Bb-CalcI-WMI/Unit4/HTML-GIF/DefiniteInt.html">Definite Integral.

Each of these physical applications considered in this unit follow a similar pattern:

• slice :
  subdivide the problem into smaller pieces
• approximate :
  sum contribution from each slice, the result should be a Riemann Sum
• limit :
  the limit of a Riemann Sum is a Definite Integral
• evaluate :
  use the Second Fundamental Theorem of Calculus and techniques for Evaluating Definite Integrals

There are many more applications of the integral.  Almost all of these can be handled by following the above general approach.

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