Tuesday March 4 Arclength and line integrals 1 Consider the curve C given by x 2 y 2 4 a Draw a sketch of this curve 2 1 2 1 1 2 1 2 b Find a parameterization of this curve x t 2 cos t y t 2 sin t 0 t 2 2 Compute the arc length of the curve C Using the above parameterization 2 q Z L p x 0 t 2 sin t y 0 t 2 cos t x 0 t 2 y 0 t 2 2 so 2 q Z L 3 Let f x y 2x y Compute R C 0 x 0 t 2 y 0 t 2 d t 0 x 0 t 2 y 0 t 2 d t 2 Z 2d t 4 0 f d s Using the above parameterization we have f x t y t 2 cos t sin t cos 2t We have ds q x 0 t 2 y 0 t 2 d t 2d t so C 2 Z Z f ds 2 cos 2t d t 0 Making the u substitution u 2t we have d u 2d t and so our integral is 4 Z 0 4 Compute 2 C x d x R R C cos t d t sin t 4 0 0 0 0 y 2 d y Using the above parameterization d x 2 sin t d t d y 2 cos t d t So Z C 2 x dx Z 0 2 2 4 cos t 2 sin t d t 8 2 Z 0 cos2 t sin t d t Making the u substitution u cos t d u sin t d t our integral becomes Z 0 8 0 u 2 d u 0 Similarly Z C 2 y dy Z 0 2 2 Z 2 4 sin t 2 cos t d t 8 sin2 t cos t d t 0 Making the u substitution u sin t d u cos t d t our integral becomes Z 8 0 0 u 2 d u 0
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