{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 5 9

# Chapter 5 9 - SECTION 5.2 THE DEFINITE lNTEGFlAL 441_2 2 2...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SECTION 5.2 THE DEFINITE lNTEGFlAL 441 _2 2 , . 2 n —2i/n A n —21/n e (e _ l) 24. From Example 3(a). we have A : hm — E e . Usmg 21 CAS. 2 e = ——— and 1:1 "#00 TL i=1 62/72. _ 1 lim — - 4‘ : 6‘2 (e2 — 1) % 0.8647. whereas the estimate from Example 3(b) using Mm n—voo n 62/" ~ 1 was 0.8632. _ b' 25. y=f(w) =c0sx. Ax: L2 = 9 andm; =0+iAzc= 1. n n n A = lim Rn : lim Zﬂmﬂﬁw : hm cos<i> .5 C25 hm R — — Céssinb Tl—>OO i:1 n—wo n—->oo [:1 TL n—toe 2n Sin<i> 2n 2n Ifb: §.thenA 23mg :1. 26. (a) 0 The diagram shows one of the n congruent triangles. AAOB. with central angle 27r/n. O is the center of the circle and AB is one of the sides of the polygon. Radius 00 is drawn so as to bisect LAOB. It ' 3 follows that 00 intersects AB at right angles and bisects AB. Thus, AAOB is divided into two right triangles with legs of length %(AB) : rsin(7r/n) and r c0s(7r/n). AAOB has area 2- %[r sin(7r/n)][r cosh/71)] = 7‘2 sin(7r/n) COS(7r/n) : érz sin(27r/n). so A” : n - area(AAOB) : énrz sin(27r/n). sin 6‘ (b) To use Equation 3.4.2. gin?) = l. we need to have the same expression in the denominator as we have in the argument of the sine function—in this case, 27r/n. . . . . ' 2 2 ' hm An : hm énr2 s1n(27r/n) : hm én'rz‘suﬁ 7r/n) ~ 1 : hm \stn/n) 7rr2. Let 6 = 21. n—>oo n—>oo n—roo 27T/7’L TL 'n—boo 27T/7L TL Then as n ~> oo. 0 ——> 0. s0 Iim mm? 2 lim smblmﬂ : (1) 7H"2 2 7W2. n—wo 27r/n 9—0 5.2 The Definite Integral \ 4 1. R4 = Z f(x1)Ax [1f : x1 is a right endpoint and Am : 0.5] i=1 : 0-5 WG-5) + W) + f(1-5) + f(2)l We) = 2 — :02] = 0.5 [1.75 + 1 + (~025) + (~2)] : 05(05) 2 0.25 The Riemann sum represents the sum of the areas of the two rectangles above the x—axis minus the sum of the areas of the two rectangles below the w—axis; that is, the net area of the rectangles with respect to the tic—axis. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online