Definite Integrals
Topics
Introduction to Definite Integrals  At A Glance:
Sample Problem
Let f (x) be the function graphed below.
We can see that
Find a constant m such that
In other words, find a constant m so that we have this:
Answer. . In order to have this equal 16, m must be 2.
We call this constant m the average value of f on [a,b]. When we take the integral of f on [a,b], you get some number. This number is like the sum of all the test scores: it's the accumulation of all the stuff.
To average that accumulation we give every x the same function value as every other x. Therefore we end up with a constant function whose integral on [a,b] is the same as the integral of f on[a,b].
The average value of f on [a,b] is a yvalue. It's the particular yvalue for which the weighted area between that yvalue and the xaxis is equal to the integral of f on [a,b]. The average value of f on [a,b] is the (weighted) height of the rectangle whose (weighted) area is equal to the integral of f on [a,b].
Let f be nonnegative for the sake of the pictures and let m be the average value of f on [a,b]. The area under m is a rectangle. Whatever area is in that rectangle but not under f must make up for the area that is under f but not part of the rectangle.
Example 1
Let f (x) = 4x^{2}. Is the average value of f on [2,2]

Example 2
Calculate the average value of on [0,3]. 
Exercise 1
Let f (x) = 3x. Find a constant m such that
Exercise 2
Let f (x) = x. Find a constant m such that
Exercise 3
Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.
 f (x) = x on [10, 0]
Exercise 4
Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.
 on [2,2]
Exercise 5
Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.
 f (x) = sin x on [ = π,π]
Exercise 6
Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.
 f (x) = 3x + 2 on [1,4]
Exercise 7
Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.
 f (x) = 2x1 on [4,2]