2 Sigma (summation) notation REVIEW In this case k is the index of summationThe lower and upper bounds of summation are 1 and 5In this case i is the index of summationThe lower and upper bounds of summation are 1 and 6
9 Area Under a Curve by Limit Definition The area under a curve can be approximated by the sum of rectangles. The figure on the left shows inscribed rectangles while the figure on the right shows circumscribed rectanglesThis gives the lower sum.This gives the upper sum.
11 Left endpoint approximation: Approximate area:(too low)
12 Right endpoint approximation: Approximate area:(too high)Averaging the right and left endpoint approximations:(closer to the actual value)
13 Approximating definite integrals: different choices for the sample points If xi* is chosen to be the left endpoint of the interval, then xi* = xi-1 and we haveIf xi* is chosen to be the right endpoint of the interval, then xi* = xi and we haveLn and Rn are called the left endpoint approximation and right endpoint approximation , respectively.
14 Can also applymidpoint approximation:choose the midpoint of the subinterval as the sample point.Approximate area:The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.
16 Approximating the Area of a Plane Region b.yyf(x) = -x2 + 5f(x) = -x2 + 55544332211xx2/ / / /2/5 4/5 6/ /To approximate the area under each curve, you must sum the area of each rectangle.See next slide
17 The right endpoints, Mi, of the intervals are 2i/5, where i = 1, 2, 3, 4, 5. The width of each rectangle is 2/5 and the height of each rectangle can be obtained by evaluating f at the right endpoint of each interval.[0, 2/5], [2/5, 4/5], [4/5, 6/5], [6/5, 8/5], [8/5, 10/5]Evaluate f(x) at the right endpoints of each of these intervals.The sum of the area of the five rectangles isHeight WidthBecause each of the five rectangles lies inside the parabolic region, you can conclude that the area of the parabolic region is greater than 6.48.
18 Approximating the Area of a Plane Region for b (con’t) b. The left endpoints of the five intervals are 2/5(i _ 1), where i = 1, 2, 3,4, 5. The width of each rectangle is 2/5, and the height of eachrectangle can be found by evaluating f at the left endpoint of eachinterval.Height WidthBecause the parabolic region lies within the union of the five rectangularregion, that the area of the parabolic region is less than 8.08.6.48 < Area of region < 8.08
20 In general for the upper sum S(n) and Lower sum s(n), you use the following for curves f(x) bound between x=a and x=b.
21 Finding Upper and Lower Sums for a Region Find the upper and lower sums for the region bounded by the graph of f(x) = x2and the x-axis between x = 0 and x = 2Solution Begin by partitioning the interval [0, 2] into n subintervals, each of lengthA.B.f(x) = x2f(x) = x244332211
25 Definition of the Area of a Region in the Plane
26 Area Under a Curve by Limit Definition If the width of each of n rectangles is x, and the height is the minimum value of f in the rectangle, f(Mi), then the area is the limit of the area of the rectangles as n This gives the lower sum.
27 Area under a curve by limit definition If the width of each of n rectangles is x, and the height is the maximum value of f in the rectangle, f(mi), then the area is the limit of the area of the rectangles as n This gives the upper sum.
28 Area under a curve by limit definition The limit as n of the Upper Sum =The limit as n of the Lower Sum =The area under the curve between x = a and x = b.
32 Example: Area under a curve by limit definition Find the area of the region bounded by the graph f(x) = 2x – x3 , the x-axis, and the vertical lines x = 0 and x = 1, as shown in the figure.
33 Area under a curve by limit definition Why is right, endpoint i/n?Suppose the interval from 0 to 1 is divided into 10 subintervals, the endpoint of the first one is 1/10, endpoint of the second one is 2/10 … so the right endpoint of the ith is i/10.