Mathematics Calculus Integrals, Lecture notes of Mathematics

Download Mathematics Calculus Integrals and more Lecture notes Mathematics in PDF only on Docsity! MATH-141 Lecture Notes 04.01.2023 TOTAL AREA To find the. orea between the graph of y= fC and the. X-axts over the. interval “Caib]: 1) Subdivide Lob) at the zeros of f- 2) Tritegrate. £ over Gach subinterval 3) Add the absolute values of the jntegrals Ex: Lek fl O= x4 and GOO = GH. For each furchon Compute : a) the de fiatte. rategral over the intewol (22). b) the area between the. goph and X-axis qver ([-2/2J. 2 2 2 z 7x am 2)4yn [Xo_ a) [& 4 dx = [a4 ‘I {4c dds [4 ad ra ~ 32 + = 32 — 3 b) The area is = square units in beth cases: The de finite integral af fC) is neg etive » but the are is positive . Beocouse orea always @ Nonnegective quantity Ex: ket fld= Sinx between X=O and %= QI Compute: a) the de. finite integral of $e) over Coan] . b) the. ore between the graph 2 f£ f (X) and the x-OKxI's over Coa) Ys Siax a T 2 a) 5 Siax dx = —-@sx 5 =- Cos2ti+ Gs O ~-144 = O 1 L) f Sia xdxy = -Cosx o ° = -CosT+ (Cos 0) TT ~-C4)+ 4 Result: = 2 The total area qr i> the sum of | Siaxd x= - COSX ay The. ak solute values of tno y T intog cols = -—Cos 27+ Cost =a -4-4 Avene |2)/+|-2]=4 + +2 Since. Areas JAl+]/Al Ex! feeder f = Siax dx Cas x £ (-# d - ~dnfulte ~dn | Cosx|ee = dn |Secx|+ ce ax 9 . ae! e*+e™ ax it [ Secxde = 7 u=Cos X du aVink dx Note: ~ tnt = dn J We Set) us e* duse™. dx Are ton ute Arc tan (e%)4 Cc Multiply and divide by (Sec x +Tanx.) = [Secx (Secx4 Tanx) dy Gecx +Tank +f Spel Sex Tonk Sec*x + Secx Tanx dy Secx+ Seext Tanx du Us &e x+ Tanx dlu=(Secx Tank+ Sct) dx =Anlulec = Lal Secx+Tanx| +o SUMMARY J tans An| Sec x| 4c J ot wdx= An | Sinx| to [ See x dx = Ln| Secxstanx | 4.0 [Case xde= = An | Cosec x+ Cotxl+c 5.6. Dernite INTEGRAL SUBSTITUTIONS cind THe AREA BeTuweeN Cueves THEOREM: Substitution in Definite Integ ral . Tf 9! is continuous on the interval Cab) ond Ff Is ContinuasS on the range. of gH) =u, than | glh? J £ (909]- g'oadx = { Ff (a) du ~ glo) 4 Z£XM: Jf 3x? (xt44 dx= ? 1 We Set | ue aL when x42-4 ., U4 =(-1),.1= 0 du= 3x* dx When Xz1= 4 u,2 44-2. a ou, = Yooa 2 a/z ~ ce T= J Jo.du = 20 |" = 2f gto) = 48 oO tt/z xf Qte Cs'Bdo~7 It/y Lowy Ve Set us Cot & du=~Csct @d& Oo t= ( uide)= “4 ]° AIL 1 = o+4 2 = ot 2 T.way =Csc 8 4=- Csc8. Cor OB Le ~lu.du = - 7 2 2 ~ csc 9 2 Te — ese 8 | TA Definite Integrals o £ Symmetric Functions THEOREM! het oa be continuous on +he Symmetric ite vrenl. Ca, a). = q Q o) If fis even , then { forde= 2. f $xadx = ° b) LL fii add > then PP p00 dx = 9 If fis even funetion the integral form -a tea is twice the ixtegral fare Oo tao. rf L is an_odd function the tecteg ral fron -a toa equals O.. Integration Ys ith respect +o y 7 L a region bounding Curves Gre described b unetionS of y 5 the bosic ferrmalor has y in place of x. 3 A= ‘ fF (y)-9 ly) Joy f(y) : right-hand Curve. F(y)- Cy) is nonnegative. 3 Cy) -Aett-hand curve- ZS: Find the area of the region in the. pest quadrant that js bounded abe by y =VK dnd below by the X-oxis Q@nd the Vue yer-2. / - jek yax = X=" Ui Yo Rod Kaa? Z5 night curve rer a = | right X tee) ty A= {em -(r)-¢ ~ 40 Pres 55 Ole) Finol Exar. Questions i) Ee ax =? TT é Us dink t= ftanu -du dus fee du t=Cosy Cosu adb= -Sinudy - (de Js = -dnltl te ~ = dn { Cos(loa) J 4c x = 7 2) f 1d © Y Gee a * _2/2 We Seta ys ctf r{ U_. clu 1 = 3. ue |" = 3(i2- ‘) 2) f ton*xdx =? — u= Cosy = Sr dx du=- Sinxdk Cos? Xx Sin x= Siar dx SintX = 1 COR? x Cos? x ars tu (-u) Cu) iF “TGs — du [de We 2% dnlul+c +lalGos Kee Heer ? fh 4 Sin? x Cost el x Si ax= 1st f[ = (2 Sinx.Cosx)™ + = Sin 2X ~ 1 [ff t-los4k gy fe af 2 [Sic = test a - [x Sax Je eae g L Cas? —Sjiax = Cos 2x Gs Int) ~Sin"X = Cos2x ime ak = Bgintx a Basic + “ substitutim|oyartt dy #) Fule xe _ 7=dx a\ +S du { LT -|- u aL du = du 4 =dX —S 4 Ve set: u(qurdu | t= 4-u" clt= -gudu