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03.04.08_Page_1

# 03.04.08_Page_1 - Section 8.1 are Length Problem Find the...

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Unformatted text preview: Section 8.1 are Length Problem: Find the length of a given curve y = ﬁx) 0H [0,5] 1) Partition [61, b] into n subintervals Diagram 1 2) Look at the typical piece Diagram 2 3) L1 = Jug — m2 + (qu )— factor = (35—35 )1 1+f(xi)——f(xz+l)2 1 1—1 (xi—JCI_1)2 Recall MVT (Mean Value Theorem) If f (x) is continuous on [a,b] and differentiable 0n (61,19) then there exists a point on c e (a, [9) such that f '(c) = w — a Apply MVT to f (x) on [xlipxl] : there is a point 99* e (xﬂgcl) such that = 1+[f'(xl*)2] (x, -xz_1)—>Ax 4) Form a sum ZHZLI =ixll+f'(xl*)2Axm L 1:1 1:1 5)lirnoi1’1+[f'(xi.*)]2Ax=I:‘/1+[f'(x)]2dx=L Held for smooth f (x) (f'(x) is continuous on (a,b)) EX Fnd the length of the curve 1 l y=—x2——lnx for 23x34 2 4 _ 4 dy 2 L-L"1+(E) dx On side ...
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