The Fundamental Theorem(s) of Calculus

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1 The Fudametal Theorem(s) of Calculus Major Problem Give a fuctio y f x over a iterval a, b with f x 0 o this iterval, fid the area uder the curve ad above the iterval o the x-axis. Furtherig the Approximatio Idea. Defie the orm of a partitio to be the largest of x, x, x 3,..., x.. The approximatios to the area should get better as gets smaller. 3. Now let 0. We hope that all of the possible approximatios to the area will coverge to the same umber o matter how the poits c k were chose. 4. If f is cotiuous o a, b, the approximatios to the area will coverge to the same umber o matter how the poits c k were chose. I this case we say that f is itegrable ad that the defiite itegral of f o a, b is b a f x x lim 0 k f c k x k If f x 0 o a, b, this value is the area uder the curve ad above the iterval o the x-axis. If f x 0 o some or all of a, b, this value ca be idetified with a siged area.

2 FudametalTheorem.b Riema Sums LeftValue f_, x_, a_, b_ : N f. x a ; MidpoitValue f_, x_, a_, b_ : N f. x a b ; RightValue f_, x_, a_, b_ : N f. x b ; Sample f_, x_, a_, b_, type_ : LeftValue f, x, a, b, MidpoitValue f, x, a, b, RightValue f, x, a, b type ; BlockCoords a_, b_, h_ : a, 0, a, h, b, h, b, 0 ; IsReal x_ : Module, If NumericQ x Im x 0, Throw "Oe or more samples are outside the domai." ; x ; FuctioF x_ : x, x^, x^3, Log x, x ^, Abs x, Cos x, Sqrt Abs x ; FuctioText "x ", "x ", "x 3 ", "log x ", " x ", " x ", "cos x ", "SqrtBox \x " ; FuctioButtos Map &, Traspose Rage Legth FuctioText, FuctioText ; RiemaBlocks f_, x_, a_, b_, _, type_ : Plot f, x, a, b, Prolog Table GrayLevel 0.8, Polygo, Lie Apped, & BlockCoords,, Sample f, x,,, type & a i b a, a i b a, i, 0,, ImagePaddig 5, 5, 5, 50, PlotLabel "estimated area " ToStrig NumberForm b a Sum IsReal Sample f, x, a i b a, a i b a, type, i, 0,, 7, 4, NumberPaddig "", "0" "\" "actual area " ToStrig NumberForm Check Chop NItegrate f, x, a, b, AccuracyGoal, I, 7, 4, NumberPaddig "", "0" Maipulate RiemaBlocks FuctioF x fff, x, 0, 5, blocks, type, blocks, 0, "umber of rectagles", 4, 70,, Appearace "Labeled", type,, "height", "left", "midpoit", 3 "right", fff, 4, "fuctio", FuctioButtos, CotrolType Setter, SaveDefiitios True

3 FudametalTheorem.b 3 umber of rectagles 70 height left midpoit right fuctio x x x 3 log x x x cos x SqrtBox \x estimated area actual area Defiite Itegrals Usig Riema Sums. Partitio a, b ito equal subitervals, each with width x k b a x 0 a, x a b a, x a b a b a,..., x a, x b. Choose c k i each subiterval. For the left Riema sum, let c k x k ; for the right Riema sum, let c k x k ; for the midpoit Riema sum, let c k x k x k. 3. Create the sum k f c k x k ad use algebraic formulas to simplify it. b 4. Compute a f x x lim k f c k x k : Example: f x x o 0, 3 x 0 0, x 3, x 6,..., x 3, x 3, x k 3 :. Choose c k i each subiterval. For the left Riema sum, let c k x k ; for the right Riema sum, let k k ; for the midpoit Riema sum, let

4 4 FudametalTheorem.b k k c k x k x k. 3. Create the sum k f c k x k ad use algebraic formulas to simplify it. b 4. Compute a f x x lim k f c k x k f x_ : x^ Left Riema Sum 3 k f k k Limit f 3 9 k, Right Riema Sum k f 3 k 3 9 Limit f 3 k 3 9 k, Fudametal Theorem of Calculus I If f is cotiuous o a, b, ad if F is ay atiderivative of f o a, b, the b a f x x F b F a. Itegrate x^, x x 3 3 x 3 3 x x 3. x

5 FudametalTheorem.b 5 Itegrate x^, x, 0, 3 9 Alterate Expressios F b F a a b f x x f b f a a b f ' x x Idea: If a quatity is give by a cotiuous fuctio f, the the quatity chages at the rate f ', ad a b f ' x x measures the accumulated chage i the quatity over a, b. Example: Itegral as Accumulatio The rate at which people eter a theater is approximated by p' t. e. t, where t 0 at oo at t 60 at :00 p.m. At oo 00 people have already arrived. How may people will be i the theather at 0 miutes before :00? p 50 p p' t t e. t t 00 Itegrate. Exp. t, t, 0, Note: the data could be give i tabular form ad require a approximatio of the itegral. Major Questio: Does a fuctio f have a atiderivative F? Fudametal Theorem of Calculus II If f is cotiuous o the iterval a, b ad x is i a, b, the x F x a f t t is a atiderivative for f x ; that is, x f t t f x d d x a F x_ : Itegrate f t, t, 0, x

6 6 FudametalTheorem.b F' x f x x x Graphical Example Cosider the fuctio f graphed below with domai 3, 4. Defie g x x f t t. Locate the relative ad absolute extrema ad iflectio poits of g. f t_ : Piecewise t, 3 t 0, Sqrt t ^, 0 t, t, t 4 Plot f t, t, 3, 4, PlotRage 4, 5,, 4, AspectRatio 3, PlotStyle Thick, Blue, AxesLabel "t", Noe g x_ : Itegrate f t, t,, x, Assumptios x Reals g 0 Π 4

7 FudametalTheorem.b 7 g' x x 3 x 0 x 0 x x x 4 0 True f x x 3 x 0 x 0 x x x 4 0 True Plot g x, x, 3, 4, PlotRage 4, 5, 4,, AspectRatio 3, PlotStyle Thick, Red, AxesLabel "x", Noe Mea Value Theorem for Itegrals If f is cotiuous o a, b, the there is a c i a, b with b a f x x f c b a, or f c b a b a f x x. The value f c is called the average value of f o a, b.

8 8 FudametalTheorem.b Example g Plot x^, x, 0, 3, Fillig Axis, PlotRage, 4, 0, 0, AspectRatio, PlotStyle Thick, Blue x^ x 0 3 Solve x^ x^ x, x 0 x 3, x 3

9 FudametalTheorem.b 9 g Plot 3, x, 0, 3, Fillig Axis, PlotRage, 4, 0, 0, AspectRatio, PlotStyle Thick, Red, FilligStyle Pik ; Show g, g Proof of Fudametal Theorem of Calculus II Let f be cotiuous o the iterval a, b, let x be i a, b, ad let F x a x f x x. The F' x lim h 0 F x h F x h = lim h 0 h a x h f x x a x f x x = lim h 0 h x x h f x x By the Mea Value Theorem for Itegrals, there is a c betwee x ad x h with x x h f x x f c x h h h f c. Thus F' x lim h 0 h x x h f x x limh 0 h h f c lim h 0 f c. As h 0, x h x ad (sice c is betwee x ad x h) c x. Sice f is cotiu

10 0 FudametalTheorem.b ous, lim h 0 f c f x ad the result is prove. Proof of Fudametal Theorem of Calculus I Let f be cotiuous o a, b, ad let F be ay atiderivative of f o a, b. The by the FTC II ad the Mea Value Theorem, F x C a x f x x for some costat C. The F b F a C a b f x x C a a f x x a b f x x.

Math 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals

Math 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals Math 1314 Lesso 16 Area ad Riema Sums ad Lesso 17 Riema Sums Usig GeoGebra; Defiite Itegrals The secod questio studied i calculus is the area questio. If a regio coforms to a kow formula from geometry,