The Definite Integral. Day 3 Riemann Sums

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1 The Defiite Itegral Day 3 Riema Sums

2 If x xy y 9, the a vertical taget exists at ( A) 2 3, 3 ( B) 3,2 3 ( C) 2 3, 3 ( D) 3,2 3 ( E) 2 3, 3 2. Use a local lieariazatio for f x 9 ta x about x 0, the approximate value of f 0.3 ( A) 3 ( B) ( C) ( D) 3.05 ( E) 3.1

2 2 1. If x xy y 9, the a vertical taget exists at ( A) 2 3, 3 ( B) 3,2 3 ( C) 2 3, 3 ( D) 3,2 3 ( E) 2 3, 3 2.

3 If x xy y 9, the a vertical taget exists at ( A) 2 3, 3 ( B) 3,2 3 ( C) 2 3, 3 ( D) 3,2 3 ( E) 2 3, 3 2. Use a local lieariazatio for f x 9 ta x about x 0, the approximate value of f 0.3 ( A) 3 ( B) ( C) ( D) 3.05 ( E) 3.1

4 HW Questios

5 Today s Learig Outcomes Evaluate itegrals usig fint Mae coectios betwee the itegral of a fuctio ad the exact area betwee a curve ad the axis Use geometric formulas to fid the area betwee a curve ad the x-axis

6 Review From Yesterday Itegral Notatio Itegrad Upper boud b Itegral Symbol f ( x) dx Lower boud a Variable Of Itegratio

7 Evaluate the Itegral Usig the Calculator 2 x 2 5 dx 2

8 Evaluate the Itegral 2 2 x dx 14 3 This is the VALUE of the INTEGRAL But, what is the Area betwee the x-axis ad the curve o the iterval [-2,2]?

9 Evaluate the Itegral 5 2x 6 dx 1

10 Evaluate the Itegral 5 1 2x 6 dx 0 The VALUE of the INTEGRAL is zero because ò 2x - 6 dx + ò 2x -6 dx = = 0 1 3

11 Evaluate the Itegral 5 1 2x 6 dx 0 What is the Area betwee the x-axis ad the fuctio o the iterval [1,5]? 3 5 ò 2x -6 dx + ò 2x -6 dx = = 8 1 3

12 Optio #2 for AREA: 5 1 2x 6 dx??? This would chage the graph at the right to be ONLY above the x-axis x 6 dx 8

13 Whe give a graph ad o fuctio... We will use Geometry Remember: The value of a INTEGRAL for regios ABOVE the x-axis is POSITIVE The value of a INTEGRAL for regios BELOW the x-axis is NEGATIVE However, the value of the AREA will always be POSITIVE. Geometric formula 1 1 remiders: 2 2 A bh A lw A r A 1 h ( b ) 1 b2 2 2

14 For f ( x) show, Fid f ( x) dx f ( x) dx f ( x) dx

15 Practice Pacet p.5

16 Thumbs up? Today s Learig Outcomes Chec Poit for Uderstadig! Evaluate itegrals usig fint Mae coectios betwee the itegral of a fuctio ad the exact area betwee a curve ad the x-axis Use geometric formulas to fid the area betwee a curve ad the x-axis

17 MORE Learig Outcomes State how rectagular approximatios ca be made more accurate. Recogize a limit statemet as the sum of ifiite rectagles. Mae coectios betwee the sum of a ifiite umber of rectagles ad itegrals. Write the sum of a ifiite umber of rectagles as a itegral.

18 Riema Sums Thus far, we have used rectagles ad trapezoids to APPROXIMATE area betwee curves ad the x-axis. It would be better if we could be more accurate i our approximatios. Braistorm with your group. Let s cosider a setch.

19 Height of th rectagle (c,f(c )) Notes Here: (c,f(c )) th rectagle c 1 c 2 c c x 0 =a x 1 x 2 x -1 x x -1 x =b (c 1,f(c 1 )) Width of th rectagle= x (c 2,f(c 2 )) Rectagles extedig form the x-axis to itersect the curve at the poits (c,f(c ))

20 If we use a ifiite umber of partitios Add up the areas of each partitio Width of each partitio EXACT lim f ( c ) x 1 Height of partitios

21 Itegral Notatio Ad sice b a We ca state that f ( x) dx represets the EXACT amout lim f ( c ) 1 x b a f ( x) dx

22 2 ways to view the limit lim f ( c ) x Number of partitios goes to 1 ifiity lim f ( c ) P 0 1 x Size of partitios goes to zero

23 Both equal the itegral lim f ( c ) lim f ( c ) P x x b a f ( x) dx

24 Formal Defiitio KNOW this Let f be a fuctio o a closed iterval [a,b], let the umbers c be chose arbitrarily i the subitervals [x -1, x ]. If there exists a umber I such that lim å P 0 =1 f (c )Dx = I o matter how P ad c s are chose, The f is itegrable o [a,b] ad I is the defiite itegral of f over [a,b].

25 Examples: Write as a itegral: 2 1. lim c x, partitioed betwee[0, 2] P 0 1

26 Examples: Write as a itegral: 1 2. lim x, partitioed betwee [1, 4] P 0 c 1

27 Examples: Write as a itegral: 2 3. lim 3( m ) 1 2m 5 x, o the iterval [ 1,3]

28 Thumbs up? MORE Learig Outcomes Chec Poit for Uderstadig! State how rectagular approximatios ca be made more accurate. Recogize a limit statemet as the sum of ifiite rectagles. Mae coectios betwee the sum of a ifiite umber of rectagles ad itegrals. Write the sum of a ifiite umber of rectagles as a itegral.

29 Pacet p.6

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,