# Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

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1 Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls

2 Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion nd n ponntil or logrithmic unction, or mpl: 3 d ln ) d

3 Dinition o Intgrtion y Prts I u nd v r dirntil unctions o, thn u dv uv v du

4 Empl u dv d uv v du Stp. Choos wht your u nd dv r. Typiclly, n ponntil trm will your dv cus it is sy to intgrt, ut ln) trm will your u, cus it is sir to dirntit. d u dv Stp. Dtrmin v nd du. To ind v, intgrt dv. To ind du, dirntit u. v d d du d d ) d d ) + C

5 Stp 3. Plug in u,v, nd du into th intgrtion y prts qution ov). du v uv u dv v d dv d du u d d Stp 4. Intgrt th scond trm, which is now vry simpl, nd writ ull solution. C C d d Stp 5. Simpliy. C C + + ) 4

6 Summry o Uss: Intgrtion y Prts For n intgrl o orm n d lt u n nd dv d For n intgrl o orm n ln ) d lt u ln ) nd dv n d

7 Mor Empls: ln ) d ln ) d 4 d ln 3 ) d

8 Solving Dinit Intgrls Until now, w wr intgrting unctions o ovr n undind intrvl o Howvr, or most mny rl world pplictions o th intgrl, w will wnt to st limits o th intrvl nd vlut our intgrl within thos limits.

9 Why? Wht dos intgrtion mn grphiclly? Assum w hv som unction ), s shown low. y is ) is And w wnt to ind th rgion undr th unction, within th intrvl, ov th is th r o th shdd rgion).

10 W cn stimt th r unr th curv y mking sris o rctngls nd inding th sum o thir rs. y is ) 3 is Assuming tht 0, nd th intrvls r quidistnt sy ), th stimtd r undr th curv is just th sum o ll rctngls, which cn writtn s: A ) + ) + )... ) 3 k k This is clld Rimnn sum R), )). W cn s tht s th numr o rctngls within th intrvl,) incrss, th mor ccurt n r w ind. So, i w tk th limit s th numr o intrvls k) gos to ininit, w cn ind n ct r.

11 So I w tk th limit o th Rimnn sum s k th numr o intrvls) pprochs ininit writtn mthmticlly low), w will gt n ct r. lim R ), ) k Tru Ar This limit is clld th intgrl o ) rom to, nd is writtn s: ) d

12 Whn is this usul? Empl: At) Ar o Amzon Rinorst Clrd Millions o Acrs) Yr Using th tools w know so r: Wht is th r clrd in 997? Just plug in tim to unction. Wht is th rt o incrs in 999? Find drivtiv nd solv. Wht is th totl r clrd twn 996 nd 00? Find dinit intgrl.

13 How do w vlut dinit intgrls? Vry simpl to solv onc you v tkn th intgrl): I F) is th intgrl solution to Thn: ) F ) + C ) d F ) F ) This is clld th Fundmntl Thorm o Clculus. Why no constnt o intgrtion? W would gt th sm constnt or F) nd F), so w would hv + C nd C, thror th constnts cncl.

14 Proprtis o Dinit Intgrls < < + ± ± c c d d d c d d d d g d d g k d k d k ) ). 5 0 ). 4 whr, ) ) ) 3. ) ) )] ) [. is constnt., ) ).

15 Empls: ) ) d d d d d d

16 Empl:. Writ mthmticl qution to dscri nrgy consumption.. Bsd on this modl, dtrmin how much nrgy ws usd twn 958 nd 98

17 Solution. First, dtrmin wht gnrl modl might st it this dt. Thn ind th ncssry prmtrs.. To ind rt t crtin tim, rmmr tht this rquirs th DERIVATIVE! To ind th totl nrgy consumption ovr n intrvl, this is th AREA undr th grph within tht intrvl INTEGRATE!

18 Empl: A smll nnotch compny s nnul proits twn th yrs o 985 nd 005 hv n modld y th ollowing qution:.5 P t ) t t + ) /. Find th rt t which proits wr incrsing in 987. Dtrmin th compny s totl proits twn 990 nd 998 Proits in millions o dollrs) Yr 985Yr 0)

19 Anothr Empl: Ar twn curvs Somtims, w will wnt to ind th r twn two unctions. Empl: A compny s nnul rvnu is shown low s th lu unction, nd thir nnul costs r shown s th rd unction. How would w ind th compny s totl proits twn tim nd tim? Proit Rvnu Cost Tim Tim

20 Solution: Proit Rvnu Cost Rvnu Cost Tim Tim Totl Proit rom Tim to Tim Tim Tim Rvnu dt Tim Tim Cost dt

21 Impropr Intgrls Impropr intgrls cn occur whn your intrvl o intgrtion gos to + or, or i th r undr th curv pprochs within st intrvl cn otn occur in grphs with vrticl symptots).

22 Impropr Intgrls ) d lim ) d ) d lim ) d

23 Empls o Impropr Intgrls: ) d Vrticl symptot 4 d d Grph shown hr y/^

24 Empls: Empl. Evlut th intgrl 0 d 3 **I you r vr trying to solv dinit intgrl nd on o your trms turns out to undind, thr my n symptot nd you will hv to instd ind tht trm s pprochs th symptot. y/^3)

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