MATH 251 Fall 2023 Section 16 2 Line Integrals Instead of integrating over an interval a b we integrate over a plane curve C given by the parametric equations or compactly we write r t x t i y t j x x t y y t a t b As in an integral in Calculus 1 we partition the parameter interval a b into n subintervals ti 1 ti of equal width and let xi x ti yi y ti The corresponding points Pi xi yi divide C into n subarcs with lengths s1 s2 sn On the ith subarc pick a point P i x i y i De nition The line integral of f along C is ZC f x y ds lim n 1 f x i y i si nXi 1 If f x y 0 RC f x y ds represents the area of one side of the fence whose base is C and whose height above the point x y is f x y ZC f x y ds Z b a f x t y t s dx dt 2 dy dt 2 dt 1 2 As parameterization of a curve is needed in evaluating line integrals let us practice parameterizing di erent curves a Parameterize the unit circle from the point 1 0 to the point 0 1 b Parameterize the unit circle from the point 0 1 to the point 1 0 c Parameterize the curve y px from 0 0 to 16 4 in two di erent ways d Parameterize the curve y px from 16 4 to 0 0 e Parameterize the line segment passing through the point 0 3 towards the point 2 7 f Parameterize the line segment passing through the point a b towards the point c d Example 1 Evaluate RC x 2y ds where C is the lower half of the unit circle x2 y2 1 going counterclockwise 3 Example 2 EvaluateRC xyds where C is the curve r t ht t4i for 0 t 1 4 Remark 2 Suppose C is a piecewise smooth curve that is C is the union of a nite number of smooth 1 We write C to indicate the curve C traversed in the reverse direction We haveRC f x y ds RC f x y ds ZC curves C1 C2 Cn where the initial point of Ci 1 is the terminal point of Ci Then f x y ds ZCn f x y ds ZC1 f x y ds ZC2 f x y ds Example 3 EvaluateRC xds where C consists of the curve y x2 from 0 0 to 1 1 followed by the line segment from 1 1 to 3 5 De nition Let C be a smooth curve de ned by parametric equations x x t y y t for a t b The line integral of f along C with respect to x is 5 The line integral of f along C with respect to y is ZC ZC a f x y dx Z b f x y dy Z b a f x t y t x0 t dt f x t y t y0 t dt It frequently happens that line integrals with respect to x and y occur together In that case we have the shorthand ZC P x y dx ZC Q x y dy ZC P x y dx Q x y dy Example 4 EvaluateRC ydx ex2dy where C is the curve y x2 from 0 0 to 1 1 6 Example 5 EvaluateRC y2dx xdy where C is the arc of the parabola x 4 y2 from 5 3 to 0 2 Line integrals in space Let f x y z be a function de ned on a smooth curve C de ned by r t hx t y t z t i for a t b The line integral of f along C is f x t y t z t s dx dt 2 ZC f x y z ds Z b Example 6 EvaluateRC xz y2 ds where C is the line segment going from 1 1 0 to 2 0 3 f x t y t z t r0 t dt dy dt 2 dz dt 2 dt Z b a a Let C be de ned by r t hx t y t z t i for a t b The line integral of f x y z along C with respect to x is 7 The line integral of f x y z along C with respect to y is The line integral of f x y z along C with respect to z is f x t y t z t x0 t dt f x t y t z t y0 t dt f x t y t z t z0 t dt a ZC f x y z dx Z b ZC f x y z dy Z b ZC f x y z dz Z b ZC a a As before we also have the shorthand P x y z dx Q x y z dy R x y z dz Example 7 EvaluateRC ydx zdy xdz where C is de ned by r t ht2 t t3i for 0 t 2 8 Previously we de ned line integrals of a scalar function f over a curve C Now we de ne line integrals of a vector eld i e a vector function F over a curve C Line integrals over vector elds Let F be a vector force eld de ned on a smooth curve C r t a t b The line integral of F along C is de ned to be ZC F dr Z b a F r t r0 t dt Motivation When F is a force eld RC F dr computes the work done by the force F on an object moving along the curve C To see why recall that given r t hx t y t i ds s dx dt 2 F r t r0 t dt Z b a dy dt 2 dt r0 t dt r0 t r0 t dt Z b F r t r0 t a Therefore ZC F dr Z b a which is exactly work F r t force z T t ds distance z Example 8 Find the work done by the force eld F x y z hcos x y z2i on an object moving along the curve C de ned by r t ht2 t 2ti for 0 t 1 Example 9 Find the work done by the force eld F x y z hx y y x zi on an object moving along the line segment from 0 1 2 to 3 4 2 9
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