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1、These icons indicate that teachers notes or useful web addresses are available in the Notes Page.,This icon indicates the slide contains activities created in Flash.These activities are not editable.,For more detailed instructions,see the Getting Started presentation.,Boardworks Ltd 2006,1 of 40,A-L
2、evel Maths:Core 3for OCR,C3.6 Differentiation and Integration 2,Contents,Boardworks Ltd 2006,2 of 40,Integrals of standard functionsReversing the chain ruleIntegration by substitutionVolumes of revolutionExamination-style question,Integrals of standard functions,Review of integration,So far,we have
3、only looked at functions that can be integrated using:,For example:,Integrate with respect to x.,The integral of,Adding 1 to the power and then dividing would lead to the meaningless expression,The only function of the form xn that cannot be integrated by this method is x1=.,This does not mean that
4、cannot be integrated.,Therefore,The integral of,We can only find the log of a positive number and so this is only true for x 0.,We can get around this by taking x to be negative.,However,does exist for x 0(but not x=0).So how do we integrate it for all possible values of x?,If x 0 so:,We can combine
5、 the integrals of for both x 0 and x 0 by using the modulus sign to give:,The integral of,Find,This is just the integral of multiplied by a constant.,Find,Definite integrals involving,It is particularly important to remember the modulus sign when evaluating definite integrals of functions involving.
6、,Find the area under the curve y=between x=3,x=1 and the x-axis,writing your answer in the form ln a.,1,3,The area is given by.,Remember that ln 1=0,units squared,Definite integrals involving,We should note that definite integrals of the form can only,be evaluated if x=0 does not lie in the interval
7、 a,b.,Integrals of standard functions,By reversing the process of differentiation we can derive the integrals of some standard functions.,These integrals should be memorized.,Integrals of standard functions,Also,if any function is multiplied by a constant k then its integral will also be multiplied
8、by the constant k.,Find.,In practice most of these steps can be left out.,Contents,Boardworks Ltd 2006,11 of 40,Integrals of standard functionsReversing the chain ruleIntegration by substitutionVolumes of revolutionExamination-style question,Reversing the chain rule,Reversing the chain rule,A very h
9、elpful technique is to recognize that a function that we are trying to integrate is of a form given by the differentiation of a composite function.This is sometimes called integration by recognition.,Let,By the chain rule:,So,It follows that for n 1,Reversing the chain rule,Suppose we want to integr
10、ate(2x+7)5 with respect to x.,If the integral is multiplied by a constant k:,Consider the derivative of y=(2x+7)6.,Using the chain rule:,=12(2x+7)5,So,Dont try to learn this formula,just try to recognize that the function you are integrating is of the form k(f(x)n f(x)and compare it to the derivativ
11、e of(f(x)n+1.,Reversing the chain rule,In general,you can integrate any linear function raised to a power using the formula:,With practice,integrals of this type can be written down directly.For example:,Reversing the chain rule,Integrate y=x(3x2+4)3 with respect to x.,Notice that the derivative of
12、3x2+4 is 6x.,Using the chain rule:,=24x(3x2+4)3,So,Lets look at some more integrals of functions of the form k(f(x)n f(x).,Now consider the derivative of y=(3x2+4)4.,Reversing the chain rule,Now consider the derivative of y=(2x3 9)3.,So,Find.,Notice that the derivative of 2x3 9 is 6x2.,Reversing the
13、 chain rule,So,Find.,Start by writing as,Now consider the derivative of y=,x2 is the derivative of(x3 1).,plus 1 is,Using the chain rule:,Reversing the chain rule for exponential functions,When we applied the chain rule to functions of the form ef(x)we obtained the following generalization:,We can r
14、everse this to integrate functions of the form k f(x)ef(x).For example:,A numerical adjustment is usually necessary.,Reversing the chain rule for exponential functions,In general,Find.,Reversing the chain rule for exponential functions,With practice,this method can be extended to cases where the exp
15、onent is not linear.For example:,Find.,Notice that the derivative of 2x2 is 4x and so the function we are integrating is of the form k f(x)ef(x).,Contents,Boardworks Ltd 2006,21 of 40,Integrals of standard functionsReversing the chain ruleIntegration by substitutionVolumes of revolutionExamination-s
16、tyle question,Integration by substitution,Integration by substitution,With practice,the technique of integration by recognition can save a lot of time.,However,when it is too difficult to use integration by recognition we can use a more formal method of reversing the chain rule called integration by
17、 substitution.,To see how this method works consider the integral,Let u=5x+2 so that,The problem now is that we cant integrate a function in u with respect to x.We therefore need to write dx in terms of du.,Integration by substitution,Now change the variable back to x:,When we used the chain rule fo
18、r differentiation we saw that we can treat informally as a fraction,so:,So if u=5x+2 and dx:,Integration by substitution,Use a suitable substitution to find.,Let u=2x2 5,Substituting u and dx into the original problem gives:,Notice that the xs cancel out.,Integration by substitution,This integral co
19、uld also have been found directly by recognition.,Now we need to change the variable back to x:,However,there are functions that can be integrated by use of a suitable substitution but not by recognition.For instance:,Use the substitution u=1 2x to find.,If u=1 2x then,Integration by substitution,Su
20、bstituting these into the original problem gives:,We also have to substitute the x so that the whole integrand is in terms of u.,Also if u=1 2x then,Integration by substitution,Changing the variable back to x gives:,Definite integration by substitution,When a definite integral is found by substituti
21、on it is easiest to rewrite the limits of integration in terms of the substituted variable.,If u=then,Using the chain rule for differentiation.,Definite integration by substitution,Rewrite the limits in terms of u:,Now we need to find x in terms of u.,u2=8 x,x=8 u2,when x=3,when x=1,The area is give
22、n by.Rewrite this in terms of u:,If u=then,Definite integration by substitution,Therefore,the required area is units squared.,Contents,Boardworks Ltd 2006,31 of 40,Integrals of standard functionsReversing the chain ruleIntegration by substitutionVolumes of revolutionExamination-style question,Volume
23、s of revolution,Volumes of revolution,Consider the area bounded by the curve y=f(x),the x-axis and x=a and x=b.,If this area is rotated 360 about the x-axis a three-dimensional shape called a solid of revolution is formed.,The volume of this solid is called its volume of revolution.,Volumes of revol
24、ution,We can calculate the volume of revolution by dividing the volume of revolution into thin slices of width x.,Volumes of revolution,The total volume of the solid is given by the sum of the volume of the slices.,The smaller x is,the closer this approximate area is to the actual area.,We can find
25、the actual area by considering the limit of this sum as x tends to 0.,This limit is represented by the following integral:,Volumes of revolution,So in general,the volume of revolution V of the solid generated by rotating the curve y=f(x)between x=a and x=b about the x-axis is:,Similarly,the volume o
26、f revolution V of the solid generated by rotating the curve x=f(y)between y=a and y=b about the y-axis is:,Volumes of revolution are usually given as multiples of.,Volumes of revolution,Find the volume of the solid formed by rotating the area between the curve y=x(2 x),the x-axis,x=0,and x=2 360 abo
27、ut the x-axis.,Volumes of revolution,Find the volume of the solid formed by rotating the area between the curve y=,the y-axis,y=1,and y=2 360 about the y-axis.,Rearranging y=gives x=.,Contents,Boardworks Ltd 2006,38 of 40,Integrals of standard functionsReversing the chain ruleIntegration by substitu
28、tionVolumes of revolutionExamination-style question,Examination-style question,Examination-style question,Use a suitable substitution to the exact value of,Let u=4x+1 so,Also u=4x+1,Rewriting the limits in terms of u:,when x=0,when x=2,u=1,u=9,Examination-style question,Rewriting the integral terms of u:,
