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MthSc108_LA_6-5_LengthSolns1

MthSc108_LA_6-5_LengthSolns1 - 6.5 6 5 Lengths of Plane...

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Unformatted text preview: 6.5 6 5 Lengths of Plane Curves g Warm Up The length of a line segment on the x‐axis: g g A line segment from P1 ( x1 , y1 ) to P2 ( x 2 , y 2 ) : A polygon: What b t th t j t Wh t about the trajectory of your d ? f dog? We could measure with some string and a ruler but ruler, a nice form la in the spirit of the form lae for area and volume formula p formulae ol me would be good. good 6.5 6 5 Lengths of Plane Curves g Warm Up To find the length of a curve we often use parametric representation. parametric Recall: ll 6.5 6 5 Lengths of Plane Curves g Agenda d 1. 1 Derive formulae . if l 2. 2 Problems: In addition to finding lengths of curves g g g given p parametrically parametrically, y, a) find the length of a curve g ) g given by y b) find the length of a curve given by ) g g y y = f( ) (x); ); x = g(y); () c) and find a curve with a given length passing through a g ) g gp g g given p point point. 3. Sh t diff 3 Short differential formula. ti l f l 6.5 6 5 Lengths of Plane Curves g 6.5 6 5 Lengths of Plane Curves g 6.5 6 5 Lengths of Plane Curves g Example: Find the length of the ellipse given parametrically by l Find h l d h f h ll ll b x = a cos t and y = bsin t d b i t, 0 ≤ t ≤ 2π 6.5 6 5 Lengths of Plane Curves g Example: Find Example Find the length of the curve x = 8cos t + 8t sin t, p g 8 it y = 8sin t + 8t cos t, 8i t 0≤ t≤π . 2 6.5 6 5 Lengths of Plane Curves g Let s Let’s take another look at the formula: What Wh t if we are given i y = f (x) explicitly? li itl ? 6.5 6 5 Lengths of Plane Curves g 6.5 6 5 Lengths of Plane Curves g Example: Fi d the length of the arc of th semicubical parabola E l Find th l th f th Find f the semicubical i bi l bl between the points (1 1) and (4 8) b h ( ) d( ) (1,1) (4,8). y2 = x 3 6.5 6 5 Lengths of Plane Curves g 4 dt, Example: Find the length of the curve x = ∫ sec t − 1 dt, l dhl h fh 0 y −π ≤ y ≤ π . π 4 4 6.5 6 5 Lengths of Plane Curves g One O more word on theory: d h ...
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