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PURE 4.8. . ~ the epposite: of differentiation. Integration oan Pe 4. The Basics a) The power rule: Ci) imerease power by 4 — then divide by it (a) stick o constant on the end +A v 3 3 eg J eae se Au /te sini res “re . 7 44 indefioe.integral “Ty ede =4 2 +e ro limits Crumbers) next 10 integral sign CC a 83) ne#d 2 a : 5 fire . 4 fe du ~ ad Joe es 3 3 nexm& Y b) constant multiples ime) du + ff dx. c) sums and differences Jleco : 9] a = | “ | oa / “4 Joris du Joede «faa = Se. ae ge 2 ry . Finding, the consort c , given: - the - a coordinate on the curve. eg! cue yy where, 4 GaP - 8, pot (4,0) a ay 3 es form ready for integration ae yr ae tare substitule 424, y= we 6+ alt ra ve (ay 3 sae tT te * 6s ardte x ce 5 ——> yrasred -5 = 6x’ -[8n 3. For Cans)” expressions . S only works when inside C) is linear akoe e a nat flor vo)" de = 4 lat)” +e okt) GLO IN aged fea’ ae - 4 Gx-4) te eg. 3 tnd 4. Further indefinite integyotion. 2 snowtiae [af [aate(2° a). bene tind fon (302-4) ST ain aeaammmmaaae een (xyBn3x? — 4)" = 48x(3x7 - 4)" b foes -aPaved fame? - oes =le-atee then. J #00) ae « Fa)te If 4 [rid] = #2), . + area. between curve Ae J. x dy and y-axis ’ positive. areo. ZA+ R, + Ral * area, bounded by —~ a curve + a Une 2 curves: ai Hdde - f gl) dx oR 7 Ae { [flx) -900)] de 4. Trnproper integrals some part of the definite integral becomes infiite A) Type 1+ one or both limits are. infinite eg J ee replace 2 with finte vale hen fake the limit as X —» o> As X>oe , 470 x B) Type 2 + the function ‘itself approaches 9° or = 0° at one or both limits a es: |, Sede > ct 4-0, the faction is undefined x as x70, 4 +02 a) 2 f Sat de replace with 4, x : [se]? then take the limitas X —> the replaced number As X—>0, > infinity 5 0% factor | 5 dy is undefined = J yer rolated obout x-axis AA = 3 > a Solid of revolution * divide volume. into slices of equal nickness » volume. Approximation of V is Za, yo x dx 5 te be 0, ten V— [ty de fhe total volume. is. divided into equally sinner "soes:, So thin’ thickness 1s practioallynO equal thickness f de for fimetions rotated b for fictions rotated b about x-axis: | V= | Ty du about Yraxis: Ve | Te dy a a 
