Physically, integrating  means finding the

  Area under the curve from a to b

Area to the left of point a
Area to the right of point b
Area above the curve from a to b

The mean value of a function f(x) from a to b is given by

 The exact value of  is most nearly

7.8036
11.807

14.034
19.611

 for 

1.9800
2.6640

2.7907
4.7520

 The area of a circle of radius a can be found by the following integral

 Velocity distribution of a fluid flow through a pipe varies along the radius, and is given by v(r).  The flow rate through the pipe of radius a is given by

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