MAT 166 Calculus for Bus/Soc Chapter 7 Notes Antiderivatives - PowerPoint PPT Presentation

6/28/2013 MAT 166 – Calculus for Bus/Soc Chapter 7 Notes Antiderivatives Integration David J. Gisch Antiderivative Antiderivative 1

6/28/2013 The Power Rule Antiderivative (Integral) • Type equation here. ��3� � � 2� � � 10��� � � 3� � �� � � 2� � �� � � 10�� � 3 � � � �� � 2 � � � �� � 10 � � � �� � 3 5 � � � � � 2 4 � � � � � 10 1 � � � � � 3 5 � � � 1 2 � � � 10� � � Antiderivative Antiderivative Example: Find ��6� � � 8� � 9��� . �� Example: Find � � � �� . 2

6/28/2013 Antiderivative Exponential Integral � � �� Example: Find � � �� . Integral Antiderivative � � � � ��� �� . Example: Find � 3

6/28/2013 Integral in Physics Integral in Physics Example: For a particular object, � � � 5� � � 4 and Position � � � � Position � � � ���� � 0 � 6 . Find the equation for velocity. Derivative Integral Velocity � � � � ���� Velocity � � � � Derivative Integral Acceleration � �� � � ���� Acceleration ���� To find � we need some information. *Note that our book uses ���� rather than ���� . Integral in Physics Example: Suppose � � � 9� � � 3 � and � 1 � 8 . Find the equation for speed. Substitution 4

6/28/2013 S ubstitution Antiderivative Example: Find � 2� � � 1 6� � �� . • The derivative has several rules but it is fairly straightforward, relatively speaking. � � � � �� � � � � 2� � � 1 � 6� � �� 5 � � • Going in reverse, finding the integral (antiderivative), is often less straightforward. � �� � 2� � � 1 � � � 5 Antiderivative Antiderivative Example: Find � 8� 4� � � 8 � �� . Example: Find � � � 3� � � 10�� . 5

6/28/2013 Antiderivative Integral S ubstitution Rules ��� Example: Find � � � ��� �� . Integral S ubstitution Rules 6

6/28/2013 Antiderivative Example: A company incurs debt at a rate of � � � � 90 � � 6 � � � 12� dollars per year, where t is the amount of time (in years) since the company began. By the fourth year the company had accumulated $16,260 in debt. a) Find the total debt function. b) How many years must pass before the total debt exceeds $40,000 ? Debt Example � � � � 90 � � 6 � � � 12� Area and Definite Integral 7

6/28/2013 Integral in Physics Integral Position � � � � Position � � � ���� Total amount of substance leaked T�t� � � Derivative Integral Integral Velocity � � � � ���� Velocity � � � � Integral Derivative Integral Rate of leak, � � Acceleration � �� � � ���� Acceleration ���� To find � we need some information. *Note that our book uses ���� rather than ���� . Area and the Integral Area Under a Curve • The area under a curve is a visual representation of the • Suppose that we have the following function and it gives integral. the rate at which water is leaking from a steel drum. • The area underneath the curve represents the amount of leakage. 8

6/28/2013 Left Hand Rule Right Hand Rule • We could estimate the area by using rectangles whose • We could estimate the area by using rectangles whose heights are determined by the value of the function on heights are determined by the value of the function on the left side. the right side. Midpoint Rule Area and the Integral • We could estimate the area by using rectangles whose • Of course we only used two rectangle but if you use more heights are determined by the value of the midpoint. it is obvious that the estimate for the area becomes more accurate. 9

6/28/2013 Area and the Integral The Definite Integral • If you use more rectangles the estimate for the area becomes more accurate. Area and the Integral Definite Integral Example: Find the area under the curve from � � 0 to � � 4 where � � � 2� and � � 4 , using the midpoint rule. � � 1 4 ��2� � � � 10

6/28/2013 Definite Integral Area and Total Change Example: Find the area under the curve from � � 0 to � � 6 where � � � 0.5� � and � � 3 , using the left hand rule. • To summarize, if we have a function giving us the rate, the area under that curve (the integral) gives us the total change. Definite Integral Definite Integral Example: The figure below shows the rate oil is leaking from a machine in a large factory (in cubic centimeters per hour) with specific rates over a 12-hour period given in the table. Approximate the total amount of leakage over a 12- hour shift. � 1 ∙ 15.2 � 1 ∙ 18 � 1 ∙ 18.8 � 1 ∙ 14.1 � ⋯ � 1 ∙ 16.6 � 1 ∙ 16.4 � 187.8 This is an estimation of the definite integral �� � � � �� � 11

6/28/2013 The Fundamental Theorem • I prefer to write it as The Fundamental Theorem of Calculus � � � � �� � ����|� � � � � � ���� � � � 2 0 � 1 2 4 � � 1 4 � � � 4 2 0 � � 128 � 2� � �� � Definite Integral Definite Integral Example: Find the following definite integral � 3 2 � � � 2 �� � � 12

6/28/2013 Definite Integral Definite Integral Example: Find the following definite integral Example: Find the following definite integral �� � 1 3� � � 3� � 1 �� 16 � 4� � � 8 � �� � � � � 0,15 , �0,450� Definite Integral • Remember the integral is the area “under” the curve (from the x-axis ). � � � � � � 4 �� � � � 4 �� � � � 4 �� � � � � � 2 � � � � � � � � � 4 �� � � 3 � 4� � � 0 � 16 3 � 32 3 � 16 � 2 � 3 � 4 2 � 0 � 3 � 4�0� � � 16 3 • The area should be positive so we take the absolute value of integrals where the graph is below the x-axis. � � � 16 � 16 � � � 4 �� � 3 3 � 13

6/28/2013 Definite Integral Definite Integral Example: Find the following definite integral � � � � � �� � � Definite Integral Page 397 #69 � � � � 40.2 � 3.50� � 0.897� � � � � � �� � � a) Find the integral of ���� over the interval �0,9� . What does this integral represent? b) Baby boomers are those born between 1945 and 1965, that is, those in the range of 4.5 to 6.5 decades in 2010. Estimate the number of baby boomers. 14

6/28/2013 Page 397 #69 The Area Between Two Curves Area Between Two Curves Area Between Two Curves Example: Find the area between the graphs below. • We can also find the area between two curves. 15

6/28/2013 Area Between Two Curves Area Between Two Curves Example: Find the area between the graphs below. • The fact that one of the curves is below the x-axis does not effect the outcome in this scenario. � �� � � �� � �� � � �� �� � � � � � � � � � � �� �� � � � � � � � �� Area Between Two Curves S avings Analysis Example: A company is considering a new manufacturing • Where it does matter is when the functions switch from process in one of its plants. The new process provides ���� � ���� to ���� � ���� or vice versa. substantial initial savings, with the savings declining with time � (in years) according to the rate-of-savings function � � � � � � � � 2� � � � 2� � � �� � � � � � � 100 � � � , � � where �′��� is in thousands of dollars per year. At the same time, the cost of operating the new process increases with time � (in years), according to the rate-of-cost function (in thousands of dollars per year) � � � � � � � 14 3 �. 16

6/28/2013 S avings Analysis Consumers’ S urplus a) For how many years will the company realize savings? • Recall that the equilibrium point is where the supply and demand curves intersect. ▫ Remember that this points represents the price where consumers will purchase the same amount of product that the manufacturer wants to sell. • However, some people would be willing to pay more than the equilibrium price for a product. The difference in these two amounts can be thought of as a “savings” b) What will the net total savings be during that period? realized by the customer. ▫ This is called the consumers’ surplus. Consumers’ S urplus Consumers’ S urplus • Here � � is the equilibrium price and � � is the equilibrium quantity. � � • The integral � � � �� would be the total amount the � consumers would be willing to pay for � � items. 17