9 Indefinite Integrals or Antiderivatives You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).
10 AntiderivativeAn antiderivative of a function f is a function F such thatEx.An antiderivative ofissince
11 Indefinite Integral The expression: read “the indefinite integral of f with respect to x,”means to find the set of all antiderivatives of f.x is called the variable of integrationIntegrandIntegral sign
12 Constant of Integration Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.NoticeRepresents every possible antiderivative of 6x.
13 Power Rule for the Indefinite Integral, Part I Ex.
14 Power Rule for the Indefinite Integral, Part II Indefinite Integral of ex and bx
15 Sum and Difference Rules Ex.Constant Multiple RuleEx.
44 Computing Area Gives the area since 2x3 is nonnegative on [0, 2]. Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph ofGives the area since 2x3 is nonnegative on [0, 2].AntiderivativeFund. Thm. of Calculus
45 Substitution for Definite Integrals Ex. CalculateNotice limits change
47 The Definite Integral As a Total If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by
48 The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by
49 A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below.What is the total distance traveled by the bee?700 feet200ft200ft200ft100ft
50 What is the displacement of the bee? 100 feet towards the hive200ft200ft-200ft-100ft
51 To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement.To find distance traveled we have to use absolute value.Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.)Or you can use your calculator to integrate the absolute value of the velocity function.
52 Displacement:Distance Traveled:velocity graphposition graphEvery AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.
53 In the linear motion equation: V(t) is a function of time.For a very small change in time, V(t) can be considered a constant.We add up all the small changes in S to get the total distance.
54 We add up all the small changes in S to get the total distance. As the number of subintervals becomes infinitely large (and the width becomes infinitely small), we have integration.
55 This same technique is used in many different real-life problems.
56 Example 5:National Potato ConsumptionThe rate of potato consumption for a particular country was:where t is the number of years since 1970 and C is in millions of bushels per year.For a small , the rate of consumption is constant.The amount consumed during that short time is
57 Example 5:National Potato ConsumptionThe amount consumed during that short time isWe add up all these small amounts to get the total consumption:From the beginning of 1972 to the end of 1973:million bushels