5.3 Definite Integrals and Riemann Sums

I. Rules for Definite Integrals

II. Mean Value Theorem for Integrals A.) If f (x) is continuous on [a, b], then there exists at least one value c in [a, b] where

B.) Proof: Using Rule #7 Since f (x) is continuous on [a, b], f must take on every value between the minimum and maximum values of f. Therefore, there will be at least one value c in [a, b] where

C.) Graphically: a b c

D.) Average Velocity Using the Mean Value Theorem for Integrals:

III. Examples Find the following:

IV. Fundamental Theorem of Calculus Part II A.) If f(x) is continuous on [a, b] and F(x) is any antiderivative of f(x) on [a, b], then…

B.) Evaluate the following definite integrals using the Fundamental Theorem of Calculus:

C.) Ex. – Evaluate the following definite integrals using the Fundamental Theorem of Calculus

1.)2.)3.)

D.) Find the average value of the following functions on the interval [1, 4].

E.) Ex.- Find the total distance traveled by a particle in rectilinear motion on [0,2] given the following velocity equation.

Total distance traveled is the area under the velocity curve bounded by the x-axis

F.) Ex.- Evaluate