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Fundamental Theorem of Calculus, Riemann Sums, Substitution Integration Methods 104003 Differential and Integral Calculus I Technion International School of Engineering 2010-11 Tutorial Summary – February 27, 2011 – Kayla Jacobs Indefinite vs. Definite Integrals A calc student upset as could be · Indefinite integral: That his antiderivative didn't agree With the one in the book The function F(x) that answers question: E'en aft one more look. Oh! Seems he forgot to write the "+ C". “What function, when differentiated, gives f(x)?” -Anonymous · Definite integral: o The number that represents the area under the curve f(x) between x=a and x=b o a and b are called the limits of integration. o Forget the +c. Now we’re calculating actual values . Fundamental Theorem of Calculus (Relationship between definite & indefinite integrals) If and f is continuous, then F is differentiable and . Important Corollary: For any function F whose derivative is f (i.e., ’ ), & % & This lets you easily calculate definite integrals! Definite Integral Properties · 0 · · whether or not , Area in [a,b] Bounded by Curve f(x) Case 1: Curve entirely above x-axis. Really easy! Area = Case 2: Curve entirely below x-axis. Easy! Area = | | = - Case 3: Curve sometimes below, sometimes above x-axis. Sort of easy! Break up into sections. Average Value The average value of function f(x) in region [a,b] is: average Calculus – Tutorial Summary – February 27,, 22001111 2 Riemann Sum Let [a,b] == cclloosseedd iinntteerrvvaall iinn tthhee ddoommaaiinn ooff ffuunnccttiioonn f PPaarrttiittiioonn [[aa,,bb]] iinnttoo nn ssuubbddiivviissiioonnss:: {{ [[xx ,x ], [x ,x ], [x ,x ], … , [x ,x ]} where a == xx < x < … < x < x = b 0 1 1 2 2 3 n-1 n 0 1 n-1 n The Riemann sum ooff ffuunnccttiioonn ff oovveerr iinntteerrvvaall [[aa,,bb]] iiss:: 1 * + , · -20 - - -/0 th where y is any value between x aanndd xx. Note (x – x ) is the length of the i ssuubbddiivviissiioonn [[xx , x]. i i-1 i i i-1 i-1 i If for all i: then… y = x S = Left Riemann sum i i-1 y = x S = Right Riemann sum i i y = (x + x )/2 S = Middle Riemann sum i i i-1 f(yi) = ( f(x ) + f(x) )/2 S = Trapezoidal Riemann sum i-1 i f(y) = maximum of f over [x , x] S = Upper Riemann sum i i-1 i f(y) = minimum of f over [x , x] S = Lower Riemann sum i i-1 i As ' ( ∞, S ccoonnvveerrggeess ttoo tthhee vvaalluuee ooff tthhee ddeeffiinniittee iinntteeggrraall of f over [a,b]: lim * 1((6 Ex: Riemann sum methods of f(x)) == x3 over interval [a, b]] == [[00,, 22]] uussiinngg 44 eeqquuaall ssuubbddiivviissiioonnss ooff 00..55 eeaacchh:: (1) Left Riemann sum: (2) Right Riemann sum: (4) Middle Riemann sum: (3) TTrraappeezzooiiddaall RRiieemmaannnn ssuumm:: Calculus – Tutorial Summary – February 27, 2011 3 Integration Method: u-substitution %7·7 %977 9 …where 7 7’ (because 7’ 7/). Notes: · This is basically derivative chain rule in reverse. · The hard part is figuring out what a good u is. · If it’s a definite integral, don’t forget to change the limits of integration! ( 7, ( 7 · If it’s an indefinite integral, don’t forget to change back to the original variable at the end, and +c. Basic Trigonometric Derivatives and Indefinite Integrals From trigonometric identities and u-substitution: Calculus – Tutorial Summary – February 27, 2011 4 Integration Method: Trigonometric Substitution for Rational Functions of Sine and Cosines To integrate a rational function of sin(x) and cos(x), try the substitution: @ tan 2 @ 2 1@? Use the following trig identities to transform the function into a rational function of t: 2 tan 2@ sin 2 ? 1tan?2 1@ 1tan?CD 1@? 2 cos 1tan?CD1@? 2 2tanC2D 2@ tan 1tan?CD1@? 2 Integration Method: Trigonometric Substitution If the integral involves… … then substitute… … and use the trig identity… E E 7 · sinG ? ? F 1sin G cos G E E 7 · tanG ? ? F 1tan G sec G E E 7 · secG ? ? F sec G1 tan G Steps: 1. Notice that the integral involves one of the terms above. 2. Substitute the appropriate u. Make sure to change the dx to a du (with relevant factor). 3. Simplify the integral using the appropriate trig identity. 4. Rewrite the new integral in terms of the original non-Ѳ variable (draw a reference right-triangle to help). 5. Solve the (hopefully now much easier) integral,
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