6. The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. The theorem ensures that if f (@ ) is continuous, a point p exists in the interval [a, b] such that the value of the function at p is equal to the average value of f (x) over [a, b]. f (p) = ba Sa f (x) da or equally So f (x) da = f (p) (b -a) A real world example of this can be shown by the average speeding ticket. How can a police officer conclude that you were in fact speeding and deserve a ticket? Well, you can determine this by using the Mean Value Theorem for Integrals by being able to determine the velocity of an object at a specific instant while observing the velocity at another instant to find the mean value of velocity. For example, you are traveling down a highway where the speed limit is 55 mph. A police officer clocks your speed at 8:05 to be 50 mph, while 5 minutes later another police officer 5 miles down the road clocks your speed at 55 mph. The second police officer pulls you over and issues you a speeding ticket. Why is that? Well according to the theorem, you traveled 5 miles in 5 minutes or 1/12hr. That is equal to 60 mph and since that is your average velocity due to the Mean Value Theorem. This mathematics enthusiast police officer can conclude that your instantaneous velocity was 60 mph at least one time within that 5 mile stretch. Shae5. The average value of a function f on an interval [a, b] can be defined as the following: f() average = b a . Sa f (x) dx. A real world application for the average value of a function can be used to determine the amount of work that is needed to stretch a spring from one length to another length. W = [ F (x) dx Specifically, utilizing Hook's Law on the amount of force acting on an object and stretching the spring to a certain distance. F(x = kx with k being the proportionality constant which depends on the parameters of the spring itself (elasticity of the material). This is measured in units of force divided by the units of distance. W = kada thus, the work produced by stretching the spring from length x = a to x =b