Calculus 220 section 6 5 Applications of the Definite Integral notes by Tim Pilachowski So far in this class we have spent chapters 1 through 5 beginning with a function that represents an amount The derivative slope of the curve gives us a rate of change In chapter 6 we are beginning with a rate of change and use the integral area under the curve to determine an amount In applications it is first important to determine the question to identify what sort of answer you are looking for Is it an amount or is it a rate of change Next identify the nature of the information you have Is it an amount or is it a rate of change Example A For a retail company the monthly marginal cost when x units have been purchased is 50 0 08x dollars Their monthly fixed costs are 700 a Determine the cost for producing x units b What is the cost of raising monthly purchases from 10 to 15 units For part a we want to know a cost the amount of money needed to purchase items The cost to purchase x units is the Cost function C x The information give is a rate of change the marginal cost is C x the derivative of the Cost function Monthly fixed costs are 700 is an amount when x 0 i e C 0 700 We can find C x from C x using the integral Answer C x 0 04 x 2 50 x 700 For part b we want to know a cost an amount The information involves a rate of change raising the number of units from 10 to 15 The numbers 10 and 15 set definite boundaries for us We need a definite integral Answer 245 Example B An object falls from a high altitude balloon with a speed of 32t feet per second How far does the object fall during the first 4 seconds How far asks for a distance which is an amount The velocity is a rate of change of distance with respect to feet change in distance time v t in is The first four seconds gives us boundaries of t 0 and t 4 second change in time Answer downward 256 feet Example C The rate at which gasoline was consumed in the U S was approximately r t 0 075t 1 7 billion gallons per year from 1964 t 0 to 1976 Find the number of gallons of gas consumed from 1964 to 1976 The equation give is a rate of change of consumption The number of gallons asked for is an amount From 1964 to 1976 gives us boundaries t 0 to t 12 We need a definite integral Answer 25 8 billion gallons All three examples above are essentially anti derivatives we had a rate of change and used the information given to determine an amount There are however other uses of integrals Consider the average of a group of x x 2 x3 K x n numbers average x 1 The average value of a function is calculated in a similar n f x1 f x 2 f x3 K f x n manner average of a function y However a continuous function n consists not of discrete values which can be added together but an infinite number of values A little algebraic manipulation will help us find a way to calculate this infinite sum First we raise to higher terms by multiplying numerator and denominator by the same number We choose the number b a f x1 f x 2 f x3 K f x n b a average of a function y n b a Rearranging using the commutative and associative properties b a 1 average of a function y f x1 f x 2 f x3 K f x n b a n The latter part of the formula should look familiar it s the Riemann sum for the area under the curve Thus b 1 1 average of a function y f x dx lim f x1 f x 2 f x3 K f x n x b a x 0 b a a Example D Example F from Lecture 6 3 revisited Blood flows through an artery fastest at the center and slowest next to the artery wall The rate of blood flow in an artery with radius 0 2 cm is described by the function v x 40 990 x 2 where v is in centimeters per second and x distance from the center in centimeters What is the average velocity of the blood flow through the artery Another application using integrals comes from the business world The text does a consumer surplus problem as Example 4 The explanation below is designed to add to your understanding of why the process works the way it does Let p price and x the number of units sold A demand curve illustrates the reality that as price of goods increases fewer people will buy and the demand goes down The function p f x will have a negative slope A predictable number of items x A will be sold if the price is set at a particular level B f A The area of the rectangle in the picture has area A B number sold times price revenue generated by sales Note that the top edge of the rectangle is the horizontal line function p f A B The area under the curve above the rectangle represents the money saved by the people who were willing to pay a higher price the consumer s surplus We already know that area under a curve can be found by integrating Thus consumer s surplus area under demand curve area of rectangle area under p f x area under p f A B A 0 f x B dx Example E Let p 15 0 5x represent a demand function where p is price in dollars and x is the number of units sold Let the level of demand 7 a What was the price b How much revenue was generated c What was the consumer s surplus Answers 11 50 80 50 12 25 You ll need Example 5 and the formula at the top of the page following it in the text to do some of the assigned homework It is similar to the continuous compounding done in section 5 2 but instead of one deposit there are a series of deposits a continuous income stream that increase the balance along with the interest If you think rt of each new deposit as an A Pe calculation where A accumulated balance the sum of balances from each deposit forms a Riemann sum i e an integral calculation From your knowledge of basic geometry you are probably already familiar with the formula for volume of a rectangular prism length times width times height If we think of that prism as laying on its side the formula looks more like volume area of cross section times length cross section cross section is circular is rectangular A width height A r2 length length The volume of a cylinder …
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