Presentation on theme: "APPLICATIONS OF INTEGRATION"— Presentation transcript:
1 APPLICATIONS OF INTEGRATION 6APPLICATIONS OF INTEGRATION
2 6.5 Average Value of a Function In this section, we will learn about: APPLICATIONS OF INTEGRATION6.5Average Valueof a FunctionIn this section, we will learn about:Applying integration to find outthe average value of a function.
3 It is easy to calculate the average value of finitely many numbers AVERAGE VALUE OF A FUNCTIONIt is easy to calculate the averagevalue of finitely many numbersy1, y2 , , yn :
4 However, how do we compute the average temperature during a day if AVERAGE VALUE OF A FUNCTIONHowever, how do we compute theaverage temperature during a day ifinfinitely many temperature readingsare possible?
5 This figure shows the graph of a temperature function T(t), where: AVERAGE VALUE OF A FUNCTIONThis figure shows the graph of atemperature function T(t), where:t is measured in hoursT in °CTave , a guess at the average temperature
6 In general, let’s try to compute the average value of a function y = f(x), a ≤ x ≤ b.
7 We start by dividing the interval [a, b] AVERAGE VALUE OF A FUNCTIONWe start by dividing the interval [a, b]into n equal subintervals, each withlength
8 Then, we choose points x1*, . . . , xn* in AVERAGE VALUE OF A FUNCTIONThen, we choose points x1*, , xn* insuccessive subintervals and calculate theaverage of the numbers f(xi*), , f(xn*):For example, if f represents a temperature function and n = 24, then we take temperature readings every hour and average them.
9 Since ∆x = (b – a) / n, we can write AVERAGE VALUE OF A FUNCTIONSince ∆x = (b – a) / n, we can writen = (b – a) / ∆x and the average valuebecomes:
10 If we let n increase, we would be AVERAGE VALUE OF A FUNCTIONIf we let n increase, we would becomputing the average value of a largenumber of closely spaced values.For example, we would be averaging temperature readings taken every minute or even every second.
11 By the definition of a definite integral, the limiting value is: AVERAGE VALUE OF A FUNCTIONBy the definition of a definite integral,the limiting value is:
12 So, we define the average value of f on the interval [a, b] as: AVERAGE VALUE OF A FUNCTIONSo, we define the average value of fon the interval [a, b] as:
13 Find the average value of the function f(x) = 1 + x2 on Example 1Find the average value ofthe function f(x) = 1 + x2 onthe interval [-1, 2].
14 AVERAGE VALUEExample 1With a = -1 and b = 2,we have:
15 If T(t) is the temperature at time t, AVERAGE VALUEIf T(t) is the temperature at time t,we might wonder if there is a specific timewhen the temperature is the same asthe average temperature.
16 For the temperature function graphed here, AVERAGE VALUEFor the temperature function graphed here,we see that there are two such times––justbefore noon and just before midnight.
17 In general, is there a number c at which AVERAGE VALUEIn general, is there a number c at whichthe value of a function f is exactly equalto the average value of the function—that is,f(c) = fave?
18 The mean value theorem for integrals states that this is true AVERAGE VALUEThe mean value theorem forintegrals states that this is truefor continuous functions.
19 If f is continuous on [a, b], then there exists MEAN VALUE THEOREMIf f is continuous on [a, b], then there existsa number c in [a, b] such thatthat is,
20 The Mean Value Theorem for Integrals is a consequence of the Mean Value Theoremfor derivatives and the Fundamental Theoremof Calculus.
21 The geometric interpretation of the Mean MEAN VALUE THEOREMThe geometric interpretation of the MeanValue Theorem for Integrals is as follows.For ‘positive’ functions f, there is a number c such that the rectangle with base [a, b] and height f(c) has the same area as the region under the graph of f from a to b.
22 Since f(x) = 1 + x2 is continuous on the MEAN VALUE THEOREMExample 2Since f(x) = 1 + x2 is continuous on theinterval [-1, 2], the Mean Value Theorem forIntegrals states there is a number c in [-1, 2]such that:
23 In this particular case, we can find c explicitly. MEAN VALUE THEOREMExample 2In this particular case, we canfind c explicitly.From Example 1, we know that fave = 2.So, the value of c satisfies f(c) = fave = 2.Therefore, 1 + c2 = 2.Thus, c2 = 1.
24 So, in this case, there happen to be MEAN VALUE THEOREMExample 2So, in this case, there happen to betwo numbers c = ±1 in the interval [-1, 2]that work in the Mean Value Theorem forIntegrals.
25 Examples 1 and 2 are illustrated here. MEAN VALUE THEOREMExamples 1 and 2 are illustratedhere.
26 Show that the average velocity of a car MEAN VALUE THEOREMExample 3Show that the average velocity of a carover a time interval [t1, t2] is the sameas the average of its velocities duringthe trip.
27 If s(t) is the displacement of the car at time t, MEAN VALUE THEOREMExample 3If s(t) is the displacement of the car at time t,then by definition, the average velocity ofthe car over the interval is:
28 On the other hand, the average value of the MEAN VALUE THEOREMExample 3On the other hand, the average value of thevelocity function on the interval is: