The Defiite Itegral Day 3 Riema Sums
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. Use a local lieariazatio for f x 9 ta x about x 0, the approximate value of f 0.3 ( A) 3 ( B) 3.005 ( C) 3.025 ( 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. Use a local lieariazatio for f x 9 ta x about x 0, the approximate value of f 0.3 ( A) 3 ( B) 3.005 ( C) 3.025 ( D) 3.05 ( E) 3.1
HW Questios
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
Review From Yesterday Itegral Notatio Itegrad Upper boud b Itegral Symbol f ( x) dx Lower boud a Variable Of Itegratio
Evaluate the Itegral Usig the Calculator 2 x 2 5 dx 2
Evaluate the Itegral 2 2 x 2 2 5 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]? 2 2 14 14 3 3
Evaluate the Itegral 5 2x 6 dx 1
Evaluate the Itegral 5 1 2x 6 dx 0 The VALUE of the INTEGRAL is zero because..... 3 5 ò 2x - 6 dx + ò 2x -6 dx = - 4+ 4 = 0 1 3
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 = -4 + 4 = 8 1 3
Optio #2 for AREA: 5 1 2x 6 dx??? This would chage the graph at the right to be ONLY above the x-axis. 5 1 2x 6 dx 8
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
For f ( x) show, Fid 2 2 2 6 6 6 f ( x) dx f ( x) dx f ( x) dx
Practice Pacet p.5
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
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.
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.
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 ))
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
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
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
Both equal the itegral lim f ( c ) lim f ( c ) P 1 0 1 x x b a f ( x) dx
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].
Examples: Write as a itegral: 2 1. lim c x, partitioed betwee[0, 2] P 0 1
Examples: Write as a itegral: 1 2. lim x, partitioed betwee [1, 4] P 0 c 1
Examples: Write as a itegral: 2 3. lim 3( m ) 1 2m 5 x, o the iterval [ 1,3]
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.
Pacet p.6