Riemann Sums - Classwork. Right rectangles Left rectangles

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1 Riemann Sums - Classwork &s a common example, this worksheet will use this problem. ;ind the area under the function f ( x) given in the picture below from x = to x = %. Ahat we are looking for is the picture on the right. Ae will look at B technicuesd right rectangles, left rectangles, midpoint rectangles and trapezoids. ;irst some common statements. Ae will use B rectangles or trapezoids in this worksheet but you are expected to learn the technicue for any number of rectangles or trapezoids. Gbviously, if we wish B rectangles and the values of x run from to %, the base of each rectangle is. Here are the B pictures of what we are looking for. Right rectangles Left rectangles Ihe height of the rectangle is on the right side. Ihe height of the rectangle is on the left side. Ihis will underestimate the area Jthis case onlyk. Ihis will overrestimate the area Jthis case onlyk. Midpoint rectangles Trapezoids Ihe height of the rectangle is in the middle. Ihis ends up both over and underestimating the area. Ihe vertical lines represent the bases of the trapezoids Ihe result is a very good approximation to the area. MasterMathMentor.com " $% " Stu Schwartz

2 Ahen we divide our picture into B rectangles, we have to find the base of each rectangle. Mn this case, since we were interested in doing this from x = to x = %, and there were B rectangles, we lucked out because the base of each rectangle is. Ihat wonnt always happen. Mn general, letns call the base = b. Oow let us define x 0, x, x, x 3... x n as the places on the x"axis where we will build our heights, where n represents the number of rectangles Jor trapezoidsk. Mn this case, x =, x =, x = 3, x = 4, x = Mn the case of left rectangles, the area will bed Mn the case of right rectangles, the area will bed A! bh + bh + bh + bh 0 3 A! b h + h + h + h 0 3 A! bh + bh + bh + bh 3 4 A! b h + h + h3 + h4 but since h = f ( x ), we can say i i A b f x0 f x f x f x3 A b f x f x f x f x! so, in the specific case above ( 3 4 )! A f f f 3 f 4 A f f 3 f 4 f 5 so in general:! n" A b f x i!! A b f x i i= 0 i= n! Ihese are called Piemann Sums. Mn the case of midpoint rectangles, you have to find the midpoint between your x 0, x, x, x 3... x n Ihe midpoint between any two x values is their sum divided by, so you will used * $( x + x )' $( x + x )' $( x + x )' $( x + x )' A! b, f & ) + f & ) + f & ) + f & )/ + % ( % ( % ( % (. * $( + ) ' $( + 3) ' $( 3 + 4) ' $( 4 + 5) '- Mn our case, A!, f & ) + f & ) + f & ) + f & )/ or A + % ( % ( % ( % (. f. 5 f. 5 f 3. 5 f 4. 5 ;or trapezoids, remember that area A = 0 height 0 ( b + b) Ihat is when the trapezoid looks like thisd b b h! Since our traezoids are on their sides, we will say A = 0 base 0 ( h + h ) So, the total area A! b %( f ( x0) + f ( x) ) + ( f ( x) + f ( x) ) + ( f ( x) + f ( x3) ) + ( f ( x3) + f ( x4) )& or, in our case A! b f ( x0) + f ( x) + f ( x) + f ( x3) + f ( x4 ) b f f f 3 f 4 f 5 % & = % & Mn general, the trapezoidal ruled A b f x0 f x f x f x3... f xn" f x % n &! MasterMathMentor.com " $L " Stu Schwartz

3 = " TetNs try oned Tet f x x 3. Ae want to find the area under the curve using U rectanglesvtrapezoids from x = to x = L. ;irst, letns draw it. Oote that the curve is completely above the axis. Mf it dips below, the method changes slightly. Ihe drawing of the curve is helpful, but not necessary. Since there are U rectangles, and we are finding the area between x = and x = L, the base is WWWWWWWWW TetNs complete the chartd i x i f ( x i ) X $ B % L S U So, the right rectangle formula gives WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW the left rectangle formula give WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW the trapezoid formula gives WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW Oote that the chart will not give you the midpoint formula. TetNs do it hered WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW Ihe calculator can generate this chart. TetNs use right rectangles. Yo to SI&I Z[MI and clear out T and T. Place your x i in T. Mt will look like thisd Oow T contains f ( x i ). Since your function is in ], use Oow, you want to sum your T list and multiply it by your base which is.%. So go to your home screen and used ]ou will find the S^M command in your TMSI M&IH menu. How do you ad`ust this for left rectanglesa WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW MasterMathMentor.com " $S " Stu Schwartz

4 Interpretation of Area. & car comes to a stop % seconds after the driver slams on the brakes. Ahile the brakes are on, the following velocities are recorded. Zstimate the total distance the car took to stop. Iime since brakes applied JsecK X $ B % belocity JftVsecK UU LX BX % X X. ]ou `ump out of an airplane. cefore your parachute opens, you fall faster and faster. ]our acceleration decreases as you fall because of air resistance. Ihe table below gives your acceleration a Jin mvsec K after t seconds. Zstimate the velocity after % seconds. t X $ B % a d.u U.X$ L.%$ %.$U B.B $.L $. eedarbrook golf course is constructing a new green. Io estimate the area A of the green, the caretaker draws parallel lines X feet apart and then measures the width of the green along that line. [etermine how many scuare feet of grass sod that must be purchased to cover the green if ak Ihe caretaker is lazy and uses midpoint rectangles to calculate the area. bk Ihe caretaker uses left rectangles to calculate the area. ck Ihe caretaker uses right rectangles to calculate the area. dk Ihe caretaker uses trapezoids to calculate the area. Aidth in feet X U %X L LX %% % $X $ MasterMathMentor.com " $U " Stu Schwartz

5 Riemann Sums - Homework ;or each problem, approximate the area under the given function using the specified number of rectanglesv trapezoids. ]ou are to do all B methods to approximate the areas. f ;unction Mnterval Oumber f ( x) = x " 3x + 4 g,bh L f x x = g,lh U x = gx,h % = sin gx,!h U $ f x B f x x Teft Pight Midpoint &nswers are belowd f Teft Pight Midpoint Irapezoids d.%.% X.B$U X.L% S.L%X U.LU S.dB S.dXd $.$B%.%B%.BB.BB% B.dSB.dSB.X$.dSB Irapezoids %. Poger decides to run a marathon. PogerNs friend ieff rides behind him on a bicycle and clocks his pace every % minutes. Poger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Ihe data ieff collected is summarized below. &ssuming that PogerNs speed is always decreasing, estimate the distance that Poger ran in ak the first half hour and bk the entire race. JIrapezoidsK Iime spent running JminK X % $X B% LX S% dx Speed JmphK X X U S X L. eoal gas is produced at a gasworks. Pollutants in the air are removed by scrubbers, which become less and less efficient as time goes on. Measurements are made at the start of each month Jalthough some months were neglectedk showing the rate at which pollutants in the gas are as follows. ^se trapezoids to estimate the total number of tons of coal removed over d months. Iime JmonthsK X $ B L S d Pate pollutants are escaping % S U X $ L X JtonsVmonthK S. ;or X " t ", a bug is crawling at a velocity v, determined by the formula v =, where t is in hours and v is + t in metersvhr. ;ind the distance that the bug crawls during this hour using X minute increments. U. &n ob`ect has zero initial velocity and a constant acceleration of 3 ft sec. eomplete the chart to find the velocity at these specified times. Ihen determine the distance traveled in B seconds. t JsecK X.%.%.% $ $.% B v ( ft sec) MasterMathMentor.com " $d " Stu Schwartz