Section 5. - Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles per hour for the second hour, nd then 55 miles per hour for the lst two hours of the trip. Wht is the totl distnce trveled? Exmple 3: A cr strts moving t time t = nd grdully speeds up over time. Its velocity t few prticulr times is shown in the tble below. Estimte how fr the cr trvels during this 2 second period. t (seconds) 4 8 2 v(t) (ft/sec) 4 7 6 Exmple 4: An object trvels with velocity v(t)=t 2 where v is in feet per second nd t is in seconds. Estimte how fr the object trveled during the first three seconds, by using left endpoints nd ) three rectngles b) six rectngles
Exmple 5: Estimte the re under the grph of f(x) = x 2 ln x on [, 5] ) using four pproximting rectngles nd right endpoints. b) using eight pproximting rectngles nd right endpoints. Definition: The re of the region tht lies under the grph of the continuous nd positive function f is the limit of the sum of the res of pproximting rectngles: Section 5. Highly Suggested Homework Problems:, 3, 5,, 3, 5 2
Section 5.2 - The Definite Integrl Definition of Definite Integrl: If f is function defined for x b, we divide the intervl [,b] into n subintervls of equl width x =(b )/n. We let x =, x n = b, nd x,x 2,...,x n be ny smple points in these subintervls, so x i lies in the ith subintervl[x i,x i ]. Then the definite integrl of f from to b is provided tht this limit exists. If it does exist, we sy tht f is integrble on[,b]. Note: Geometriclly, the definite integrl represents the cumultive sum of the signed res between the grph of f(x) nd the x-xis from x = to x = b, where res bove the x-xis re counted positively nd res below the x-xis re counted negtively. Exmple : Approximte 3 smple points to be the midpoints of ech subintervl. (2x 2 x 2) dx by using the Riemnn sum with 6 equl subintervls, tking the 3
Exmple 2: Clculte the following given f(x) below Are of A is.3 Are of B is 2.5 Are of C is 2 Are of D is.75 f(x) e A g B C h D j ) g f(x) dx b) e f(x) dx c) j e f(x) dx Exmple 3: Evlute the integrl by interpreting it in terms of res. ) 3 8 dx b) 5 x 2 dx c) 2 4 (2x+5) dx 4
Properties of Definite Integrls: If f nd g re continuous functions, then. 2. 3. 4. 5. 6. b b b b b c dx= f(x) dx= f(x) dx= c f(x) dx= [ f(x)±g(x)] dx= f(x) dx= Exmple 4: Given ) 4 4 (4x 2 9x) dx x dx=7.5, 4 x 2 dx=2, nd 5 4 x 2 dx=6/3, clculte the following: b) 5 ( 4x 2 ) dx Section 5.2 Highly Suggested Homework Problems:, 3, 5, 9, 3, 33, 37, 4, 43, 49 5
Section 5.3 - Evluting Definite Integrls Evlution Theorem If f is continuous function on[,b], nd F is ny ntiderivtive of f, then Exmple : Evlute the following: ) 3 2 ( x 2 + 4) dx b) 4 (4t+ t) dt c) 5 (e x + cosx) dx d) 2 (2x+3) 2 dx 6
Indefinite Integrls The nottion f(x)dx is trditionlly used for generl ntiderivtive of f nd is clled n indefinite integrl. Thus, f(x)dx=f(x) mens F (x)= f(x) Note: We connect the two types of integrls by the Evlution Theorem, Tble of Indefinite Integrls [ f(x)±g(x)]dx= f(x)dx± b ( )] b f(x)dx= f(x)dx = F(b) F() g(x)dx c f(x)dx=c x n dx= xn+ n+ +C(n ) x dx=ln x +C e x dx=e x +C sinxdx= cosx+c sec 2 xdx=tnx+c secxtnxdx=secx+c Exmple 2: Evlute the following: x 2 + x+ dx x x dx= x ln +C f(x)dx cosxdx=sinx+c csc 2 xdx= cotx+c cscxcotxdx= cscx+c 7
Estimting Definite Integrls on the Clcultor: You cn estimte the vlue of the definite integrl by using the following commnd from your homescreen: b f(x) dx Net Chnge Theorem The integrl of rte of chnge is the net chnge of the originl function: b f (x)dx= Exmple 3: A honeybee popultion strts with bees nd increses t rte of n (t) bees per week. Wht does + 5 n (t)dt represent? Exmple 4: A forest fire covers 2 cres t time t =. The fire is growing t rte of 8 t cres per hour, where t is in hours. How mny cres re covered 24 hours lter? Exmple 5: The velocity function (in meters per second) is given for prticle moving long line. Find () the displcement nd (b) the distnce trveled by the prticle during the given time intervl. v(t)=t 2 2t 8, t 6 Section 5.3 Highly Suggested Homework Problems: 3, 7, 9,, 3, 5, 7, 2, 43, 49, 5, 53, 59, 6 8
Section 5.4 - The Fundmentl Theorem of Clculus Exmple : Let g(x)= x f(t)dt where f is the function whose grph is shown. 2 2 2 3 4 5 ) Evlute g(), g(2), nd g(4). b) On wht intervl(s) is g incresing? decresing? c) Where does g hve n bsolute mximum vlue? Absolute minimum vlue? Exmple 2: If g(x)= x t 3 dt, ) Find formul for g(x). b) Wht does your nswer represent? c) Find g (x). 9
Fundmentl Theorem of Clculus Suppose f is continuous on[, b].. If g(x)= 2. x Alternte Nottion: d dx b f(t)dt, then g is n ntiderivtive of f, tht is g (x)= f(x) for <x<b. x f(t)dt = f(x) f(x)dx=f(b) F(), where F is ny ntiderivtive of f, tht is F = f Exmple 3: Use Prt I of the Fundmentl Theorem of Clculus to find the derivtive of the following functions. ) g(x)= x 3 e t2 t dt b) h(x)= x 2 +r 3 dr c) g(x)= cosx sinx (+v 2 ) dv
Exmple 4: Let g(x)= x f(t)dt, where f is the function whose grph is shown. ) At wht vlues of x do the locl mximum nd minimum vlues of g occur? b) Where does g ttin its bsolute mximum vlue? c) On wht intervls is g concve downwrd? Section 5.4 Highly Suggested Homework Problems: 3, 5, 7, 9,, 5, 7, 9, 25
Section 5.5 - The Substitution Rule Recll tht d dx ( f[g(x)])= Thus, Exmple : Wht is e x3 3x 2 dx? Generl Indefinite Integrl Formuls. e f(x) f (x) dx= 2. 3. 4. [ f(x)] n f (x) dx= f(x) f (x) dx= cos[ f(x)] f (x) dx= Integrtion by u-substitution. Select u (look for function where you normlly hve x) 2. Tke the derivtive of u using du dx nottion. 3. Bring dx to the right hnd side. 4. Bring ny constnt multiples to the left-hnd side. 5. Substitute to replce ll terms with x s. 6. Integrte with u s. 7. Return x s into the problem. 2
Exmple 2: Let s look gin t e x3 3x 2 dx Exmple 3: Evlute the following: ) 7(8x+3) dx b) 2x 2 4 x 3 + 2 dx c) 3(x 3 + ) (3x 4 + 2x) 7 dx d) 2x 3x 2 + 5 dx 3
e) x 5 7x 2 + dx f) (2x 6 e x7 + ) dx g) e x e x e x dx + e x h) x x+2 dx i) x(x 5 + ) 2 dx 4
j) 2 (2x+3) 6 dx k) π xcos(x 2 )dx Exmple 4: An oil storge tnk ruptures t time t = nd oil leks from the tnk t rte of r(t)= e.t liters per minute. How much oil leks out during the first hour? Section 5.5 Highly Suggested Homework Problems:, 3, 5, 7, 9, 3, 5, 7, 9, 2, 23, 25, 29, 3, 4, 45, 47, 5, 53, 55, 63, 67 5