Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin

Chapter 05: Integration 5.1 Area and Estimating with Finite Sums 5.2 Sigma Notation and Limits of Finite Sums 5.3 The Definite Integral 5.4 The Fundamental Theorem of Calculus 5.5 Indefinite Integrals and the Substitution Method 5.6 Substitution and Area Between Curves

Chapter 05 Overview Many difficult quantities can be calculated by considering the sum of approximations of small portions of the whole quantity. If these approximations are in the form of a product and if the result improves when more and smaller portions are considered then the exact quantity can be calculated as a limit. This procedure is called integration.

05.01: Area and Estimating with Finite Sums 1 Left/Midpoint/Right approximations to the Area under a curve (upper/lower approximations can be ambiguous). Example 1, Lab: TI AREA Program Approximating the Distance an object has traveled. Example 2 Distance verses Displacement. Example 3 Approximating the Average Value of a nonnegative function. Example 4

05.02: Sigma Notation and Limits of Finite Sums 1 Sigma Notation (index variables, limits of the index variable, a formula that depends on the index variable). Example 1 Re-indexing a summation. Example 2 The algebra of summations. Examples 3 & 4 Limits of summations. Example 5 A Riemann Sum over an interval [a,b] on the x-axis is the summation where c k is any x-value in the k th subinterval of [a,b] and Δx = (b - a)/n. Example 6

05.03: The Definite Integral 1 The limit of a Riemann Sum is called the Definite Integral. Calculation of the Definite Integral is often very difficult. Rules for Definite Integrals. Examples 2 & 3 The Definition of Area:. Example 4 The importance of Definitions in mathematics (unit circle trigonometry, area …). The Average Value or Mean of a function. Example 5

05.04 The Fundamental Theorem of Calculus 1 The Mean Value Theorem for Definite Integrals: If f is continuous on [a,b], then at some point c in [a,b],. MVT: There is at least one point in (a,b) where the curve will attain its average value. The MVT answers the question of whether there is a rectangle with base [a,b] that has the same area as the area under the curve.

05.04 The Fundamental Theorem of Calculus 2 Specified Constants, Unspecified Constants and Variables (y = π x + b). Dummy variables and Accumulation functions (graphing Accumulation functions on a TI calculator). Fundamental Theorems: –Arithmetic: Every natural number is either a prime number or can be written as a unique product of primes. –Algebra: Every n th degree polynomial has n complex roots (it can be written as a unique product of first degree complex factors).

05.04 The Fundamental Theorem of Calculus 3 The Fundamental Theorem of Calculus (the antiderivative part): –An antiderivative of any continuous function f [t] can be constructed as the limit of the Riemann Sum represented by the Accumulation Function. This limit may not be possible to find. –The rate of accumulation (slope of the tangent line to the accumulation function) of the function f [t] at some value of x is equal to the value of the function at that value of x. Integration can be seen as the accumulation of a quantity which is described by its rate of change. –Differentiation and Integration are inverses of each other. Example 2

05.04 The Fundamental Theorem of Calculus 4 The Fundamental Theorem of Calculus (the evaluation part): –Definite Integrals can be computed using the antiderivative if it can be found instead of using the limit process. Examples 3, 5, 7 & 8

05.05: Indefinite Integrals and the Substitution Method 1 The integral symbol represents a family of antiderivatives but it can also be considered as representing a Riemann Sum and manipulated algebraically. Inverting the Power Rule. Equation (1) The Chain Rule can sometimes be inverted by using a substitution u[x] for part of the integrand so that it becomes (f [u])(u '[x]). If the antiderivative of f [u] can found then F[x] is obtained because the derivative of F[u] is f [u])(u '[ x]. Examples 1 – 8 Antiderivatives can be found by Guessing and Checking. The identity for finding the antiderivatives of sin 2 x and cos 2 x is used frequently and needs to be memorized. Example 9 A useful average of a periodic function. Example 10

05.06: Substitution and Area Between Curves 1 Substitution in Definite Integrals changes the limits as well as the integrand. Changing the limits as a result of a substitution may be avoided by finding the indefinite integral of the integrand first and then using the original limits. Examples 1 & 2 Finding the value of Definite Integrals by considering symmetry (even/odd functions). Example 3 The value of a Definite Integral is not always the same as the area trapped by the integrand curve. Finding the area between two curves. Examples 4 – 6 Finding area using vertical intervals (Δy). Example 7