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) k = ( + 1)( + 1) 6 ( ( + 1) k = ) PRACTICE PROBLEMS: For problems 1-5, evaluate. 1... 4 k 100 6 (j 1) j= 45 i i= 1 10 1
4. 5. 5 ( 1) k k=0 0 5 1 ( π ) si k For problems 6-8, use the summatio formulas at the top of page 1 to evaluate the give sum. 6. 7. 8. 100 (k 5) 14,650 5 [k(k 1)(k + 1)] 105,00 10 k= (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!) 8,08; Video Solutio: https://www.youtube.com/watch?v=cq08chq0wly For problems 9-1, 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() + 4() + 4(4) + + 4(0) 0 4k 10. 6 + 9 1 + 6 1 ( 1) k+1 k
11. 1 + + 5 + 7 + + 1 10 k=0 (k + 1) 1. + 4 + 8 + 16 + + 56 8 k For problems 1-15, express the give summatio i closed form. 1. j=1 j + 1 14. 1 k ( 1) 4 ( ) 1 15. k 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!) 1 ( + 1)( + 1) 6 + 1 16. Cosider f(x) = x + 1. (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?
A 118; It is a overestimate. (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? A 46; It is 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? A 76; By ispectio, it is hard to judge whether this is a overestimate or a uderestimate. I fact, i a future sectio, you will be able to show that the exact area is 78. 4
17. Let f(x) = l x. (a) Sketch the graph of f(x). Label all asymptotes ad itercepts with the coordiate axes. y 1 0 1 4 5 x K1 K (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? A e l (e) + e l (e) + e l (4e) + e l (5e).89 This is a overestimate. (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? 5
A e+e l (e)+e l (e)+e l (4e) 19.51 This is a uderestimate. 18. Let f(x) = x + 1. 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. x = 5 (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 1 + 5 k = x 1 + 5 () k = x 1 + 5 ()......... k = 1 x 1 1 + 5 ( 1) k = x 1 + 5 () = 6 (c) Fill i the blak: A closed formula for the right edpoits foud i the table above is x k = 1 + 5 (k), for k = 1,,..., 1,. 6
(d) Determie f(x k), the height of the k th rectagle. ( 1 + 5 k ) + 1 (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: A f(x 1) x + f(x ) 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(x k) x = 1 + 5 ) k 5 5( + 1) 15( + 1)( + 1) + 1] = 10 + + 6 (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 + 1 o the iterval [1, 6]. 0 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] 0; Detailed Solutio: Here (b) f(x) = x o [, 5] 1; Detailed Solutio: Here (c) f(x) = x 1 o [0, ] 0. 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] 0 7
(b) f(x) = x o [, 5] 1 (c) f(x) = x 1 o [0, ] 1. 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] 0 (b) f(x) = x 1 o [, 5]. 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. ( cos π + π ) π k. 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. 8
(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 π radias, as show i the diagram above for several differet values of. Show that A = 1 ( ) π r si. We begi by examiig oe of the triagles, pictured below. The base of the triagle has a legth of r. Ad, the height of the triagle is r si θ, where θ is the cetral agle, π. Thus, the area of oe triagle is: A = 1 ( )) π (r (r) si = 1 ( ) π r si But, the polygo is composed of such triagles. So, the area of a regular -sided polygo iscribed i the circle of radius r is: A = 1 ( ) π r 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. lim A = πr 9