67 Iefiite Itegral Lecture iscusse atierivatives. I this sectio, we itrouce ew otatio a vocabulary. The otatio f iicates the geeral form of the atierivative of f a is calle the iefiite itegral. From the efiitio above, we see that the otatio f trasforms f ito the atierivative. I Lecture, this process was calle atiifferetiatio. Here, we call it itegratio. The symbol is calle a itegral symbol. We recogize the otatio as a ifferetial (the chage i ), but i the cotet of itegratio iicates that is the variable to be itegrate. Suppose we wat to itegrate a costat like 5, that is, trasform 5 ito a fuctio (or set of fuctios) whose erivative equals 5. We call this itegratio a iicate it by writig 5, but we ee or t or some iicatio of the variable. If we write 5 the is the The otatio f eotes the iefiite itegral. The statemet f F meas F ' f. variable a 5 5 C. If we write 5t the t is the variable a 5t 5t C. As a operator, the itegral operates o fuctios with respect to a variable, so the argumet of itegratio is a fuctio calle the itegra. Moreover, itegratio trasforms the itegra ito a fuctio (or set of fuctios) whose erivative equals the itegra. Like ifferetiatio, itegratio is a liear operatio as efie below. If f a g are fuctios with atierivatives over some iterval I a c is a scalar, the f g f g cf c f Thikig of our rules for ifferetiatio i reverse, we ca write some rules for itegratio. For istace, the power rule of ifferetiatio gives us the rule below. C where. e e Specific ifferetiatio facts such as covice us (whe we thik of them i reverse) that the followig iefiite itegrals are true over some appropriate iterval.
68 Table of Iefiite Itegrals. C (where ). 3. e e C a a C l a 4. l C 5. si cos C 6. cos si C 7. sec ta C 8. csc cot C 9. sec ta sec C 0. csc cot csc C. ta C. si C We ca verify the rules give i the table above by ifferetiatig the fuctio o the right sie a obtaiig the itegra.
69 For eample, let's verify the first rule i the table. F C F ' C F ' 0 F ' F ' F ' Now, let s verify the fifth rule i the table. If, the F cos C F ' C F ' cos 0 F ' si cos C Eample Eercise 4 Evaluate 4.. Apply this rule as below. 5 5 4 4 4 4 5 4 C 5 C Eample Eercise Evaluate 7 sec e si. Apply the rules from the table of iefiite itegrals. 7 sec e si ta ta e 7 l cos C
70 Practice Problems i Calculus: Cocepts a Cotets by James Stewart st e. problem set: e. problem set: 3r e. problem set: Sectio 5.3 #39 45 o Sectio 5.3 #4 45 o Sectio 5.3 #37 4 o Practice Problems i Calculus: Early Trasceetals by Briggs a Cochra st e. problem set: Sectio 4.8 #9-33 o, #73 # The otatio f Possible Eam Problems refers to which of the followig objects? a) a umber b) a itegra c) a set of fuctios ) a erivative Aswer: c) a set of fuctios # If f tt F t, the what special property oes F t possess? Aswer: Ft possess the property that F ' t f t. #3 Fi. 5 4 Aswer: 5 si 5 C
7 Applicatio Eercise Differetial equatios appear i several scieces icluig physics a ecoomics. Itegratio sometimes solves ifferetial equatios of the form below. y f The fuctio (or fuctios) y whose erivative equals f solves the equatio above. Hece, the geeral solutio of the ifferetial equatio y f is the iefiite itegral below. y f Use the iscussio above to solve the ifferetial equatio below. Fi a particular solutio give that y. y 6 Applicatio Eercise Let s cocer ourselves with the istace travelle by a object roppe ear the earth s surface at t 0 assumig air resistace is egligible. I the absece of air resistace, all fallig boies accelerate at the same rate. Close to the surface of the earth the gravitatioal acceleratio of a fallig boy has the costat value g 3ft sec. Recall that the fuctio escribig the acceleratio of a movig object equals the erivative of the fuctio escribig the object s velocity, vt. Recall also that the fuctio escribig the velocity of a movig object equals the erivative of the fuctio givig the object s positio (where positio is a istace relative to some s t. arbitrary poit). Let st eote istace the fallig object has falle. Fi
7 Practice Problems Perform the iicate itegratio. # #3 8 # 3 #4 e 4 3 4 6 #5 5 5 3 #6 cos si #7 sec csc #8 3 #9 7 #0 # # sec ta 4 4 Verify by ifferetiatio that the followig formulas are correct for all real umbers C a b. si 4si #3 #4 l C l C #5 b b b C # #3 5 5 3 C #5 C 5 9 C 9 #7 ta cot C #9 l C # arcsi C