Area As A Limit & Sigma Notation

Transcription

1 Area As A Limit & Sigma Notatio SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should referece Chapter 5.4 of the recommeded textbook (or the equivalet chapter i your alterative textbook/olie resource) ad your lecture otes. EXPECTED SKILLS: Uderstad ad kow how to evaluate the summatio (sigma) otatio. Be able to use the summatio operatio s basic properties ad formulas. (You do ot eed to memorize the Useful Formulas listed below; if they are eeded, they will be provided to you). Kow how to deote the approximate area uder a curve ad over a iterval as a sum, ad be able to fid the exact area usig a limit of the approximatio. Be able to fid the et siged area betwee the graph of a fuctio ad the x-axis o a iterval usig a limit. USEFUL FORMULAS k = ( + 1) 2 k 2 = ( + 1)(2 + 1) 6 ( ( + 1) k = 2 ) 2 PRACTICE PROBLEMS: For problems 1-5, evaluate k 6 (j 1) j=2 2i i= 1 5 ( 1) k k=0 1

2 5. 5 ( π ) si 2 k For problems 6-8, use the summatio formulas at the top of page 1 to evaluate the give sum k= (k 5) [k(k 1)(k + 1)] (k + 7) (CAUTION: I problem 8, the lower idex is ot 1; so, the summatio formulas at the top of page 1 do ot immediately apply!) For problems 9-12, write the give expressio i sigma otatio. Do ot evaluate the sum. (For each, there are may differet ways to write the expressio i sigma otatio; the aswer key illustrates oe such way for each.)) 9. 4(1) + 4(2) + 4() + 4(4) + + 4(20) For problems 1-15, express the give summatio i closed form. 1. j=1 j k ( ) 1 k2 k=0 (CAUTION: I problem 15, the lower limit is ot 1; so the summatio formulas at the top of page 1 do ot immediately apply!) 2

3 16. Cosider f(x) = x (a) Estimate the area uder the graph of f(x) o the iterval [0, 6] usig rectagles of equal width ad right edpoits, as i the diagram below. Is your estimate a overestimate or a uderestimate? (b) Estimate the area uder the graph of f(x) o the iterval [0, 6] usig rectagles of equal width ad left edpoits, as i the diagram below. Is your estimate a overestimate or a uderestimate? (c) Estimate the area uder the graph of f(x) o the iterval [0, 6] usig rectagles of equal width ad midpoits, as i the diagram below. Is your estimate a overestimate or a uderestimate?

4 17. Let f(x) = l x. (a) Sketch the graph of f(x). Label all asymptotes ad itercepts with the coordiate axes. (b) Sketch the graph of f(x) o the iterval [e, 5e]. Divide the iterval ito 4 subitervals of equal width. O each subiterval, sketch a rectagle usig the fuctio value at the right edpoit as the height of the rectagle o that subiterval. Estimate the area betwee the graph of f(x) ad the x-axis o the iterval [e, 5e] usig the 4 rectagles that you sketched. Is your estimate a overestimate or a uderestimate? (c) Sketch the graph of f(x) o the iterval [e, 5e]. Divide the iterval ito 4 subitervals of equal width. O each subiterval, sketch a rectagle usig the fuctio value at the left edpoit as the height of the rectagle o that subiterval. Estimate the area betwee the graph of f(x) ad the x-axis o the iterval [e, 5e] usig the 4 rectagles that you sketched. Is your estimate a overestimate or a uderestimate? 18. Let f(x) = x By the ed of this problem, you will have computed the exact area uder the graph of f(x) o the iterval [1, 6]. (a) Fid the x which is ecessary to divide [1, 6] ito subitervals of equal width. (b) I each of the subitervals of equal width, pick x k to be the right edpoit. Fill i the followig table: Subiterval Number Right Edpoit Number Right Edpoit of Subiterval k = 1 x 1 k = 2 x 2 k = x k = 1 k = x 1 x (c) Fill i the blak: A closed formula for the right edpoits foud i the table above is x k =, for k = 1, 2,..., 1,. (d) Determie f(x k), the height of the k th rectagle. (e) The right edpoit approximatio of the area uder the graph of f(x) o the iterval [1, 6] usig rectagles of equal width is: 4

5 A f(x 1) x + f(x 2) x f(x 1) x + f(x ) x = f(x k) x Usig the appropriate formulas from the top of page 1, express the right edpoit approximatio i closed form. (f) Repeatig over fier ad fier partitios is equivalet to the umber of subitervals,, approachig ifiity. Usig this iformatio, compute the exact area uder the graph of f(x) = x o the iterval [1, 6]. 19. For each of the followig, use sigma otatio ad the appropriate summatio formulas to evaluate the et siged area betwee the graph of f(x) ad the x-axis o the give iterval. Let x k be the right edpoit of the k th subiterval (where all subitervals have equal width). (a) f(x) = x o [1, 5] (b) f(x) = x2 o [2, 5] (c) f(x) = x 1 o [0, 2] 20. For each of the followig, use sigma otatio ad the appropriate summatio formulas to evaluate the et siged area betwee the graph of f(x) ad the x-axis o the give iterval. Let x k be the left edpoit of the k th subiterval (where all subitervals have equal width). (a) f(x) = x o [1, 5] (b) f(x) = x2 o [2, 5] (c) f(x) = x 1 o [0, 2] 21. For each of the followig, use sigma otatio ad the appropriate summatio formulas to evaluate the et siged area betwee the graph of f(x) ad the x-axis o the give iterval. Let x k be the midpoit of the k th subiterval (where all subitervals have equal width). (a) f(x) = x o [1, 5] (b) f(x) = x2 o [2, 5] 22. Use sigma otatio ad the appropriate summatio formulas to formulate a expressio which represets the et siged area betwee the graph of f(x) = cos x ad the x-axis o the iterval [ π, π]. Let x k be the right edpoit of the k th subiterval (where all subitervals have equal width). DO NOT EVALUATE YOUR EXPRESSION. 5

6 2. A Regular Polygo is a polygo that is equiagular (all agles are equal i measure) ad equilateral (all sides have the same legth). The diagram below shows several regular polygos iscribed withi a circle of radius r. (a) Let A be the area of a regular -sided polygo iscribed withi a circle of radius r. Divide the polygo ito cogruet triagles each with a cetral agle of 2π radias, as show i the diagram above for several differet values of. Show that A = 1 ( ) 2π 2 r2 si. (b) What ca you coclude about the area of the -sided polygo as the umber of sides of the polygo,, approaches ifiity? I other words, compute lim A. 6