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Calculus with Applications

by: Sienna Bradtke III

Calculus with Applications MATH 11

Sienna Bradtke III
GPA 3.69


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This 3 page Class Notes was uploaded by Sienna Bradtke III on Monday September 7, 2015. The Class Notes belongs to MATH 11 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/182213/math-11-university-of-california-santa-cruz in Mathematics (M) at University of California - Santa Cruz.


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Date Created: 09/07/15
Math 11B Spring 2008 Handout 15 Copyright by Hongyun Wang UCSC Two examples of arclength Optional Example 1 Length ofy x2 from x 0 to x 1 f x 2x gt LH1f39x 2dxjJ14x2 dx 1 Use substitution x Etan9 1 2 dx 5 sec 9619 11 4x2 1tan2 9 sec9 gt IJ1 4x2 dx Ise039d9 Now we derive J secs 9 d9 sec 9tan9 logl sec 9 tan9 I c First we have 1 Isec9 d9 Male cos9 Jcosz 9 9 1 J ds1n9 I du Use substitution u sin9 1 10 1u 1 d du 1r1 11 2 Lu l u1J 71logu 1 logu 1c Math 1 1 B 0 Substitute u sin9 log 1 sin9 0059 1og seeetane c Using integration by parts on 15603 9 d9 we have jeee3ede jeeeesee2ede Isecltegtdtanltegt sec9tan9 jtanedseee sec9tan9 jeeeetan2ede sec9tan9 jseee3 019 jseeede gt jsee3 9519 sec9tan9 jseeede sec9tan910gsec9tan9I1 6 Substituting tan9 2x and seee m into the above we obtain JI14x2 dx 2xxll4x2 10g 2xxll4x2 c 1 gt L 14x2dx 0 2xquot14x2 10g2xxll4x2 0 1 Z2J log2J5 Math 1 1 B Example 2 Length ofy 10gx from x 1 to x 2 f x i Li UK z fgykji lw Use substitution u 4x2 1 du x21 Z Z Jx 1 x 1 x gt d d J x x J x2 x21x Z md J1 uzlIdu 1 Iu 1u1d 1 F 1 1 1 51 Lm m 1quot uloqu 1 10gu1c ullog u l 2 u1 u 1 c 1 11 1 u0 7 6 Substituting u sz 1 into the above we obtain Z IXTdeJxZ 1 10g x21 1 c


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