Evaluate the line integral, where C is the given curve C xy ds C: x=t2,y=2t,0≤t≤5

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Evaluate the line integral, where C is the given curve  C xy ds  C: x=t2,y=2t,0≤t≤5

Dora · Accepted Answer

Integral solution:

Jeffrey Jordon · Accepted Answer

To evaluate the line integral ∫Cxyds, where C is the curve C:x=t2,y=2t,0≤t≤5, we first need to parameterize the curve in terms of a single parameter.We can parameterize the curve by letting x=t2 and y=2t, which gives us the parametric equations x=t2 and y=2t, with 0≤t≤5.Next, we need to express ds in terms of dt. We haveds=(dxdt)2+(dydt)2dt.Substituting x=t2 and y=2t into this equation, we getds=(2t)2+(2)2dt=4t2+4dt=2t2+1dt.Now we can evaluate the line integral:∫Cxyds=∫05(t2)(2t)(2t2+1)dt=4∫05t3t2+1dt.To evaluate this integral, we can use the substitution u=t2+1, which gives us du=2tdt. Substituting this into the integral, we get∫05t3t2+1dt=12∫126(u−1)3/2du.Using the power rule for integration, we can evaluate this integral to get12∫126(u−1)3/2du=12·25(u−1)5/2|126=15(26677−82−5).Therefore, the value of the line integral is 15(26677−82−5).

RizerMix · Accepted Answer

The given line integral can be expressed as:∫Cxy&amp;nbsp;dswhere C is the curve defined by x=t2, y=2t, and 0≤t≤5.To evaluate the line integral, we need to parameterize the curve in terms of a single variable. We can choose t as the parameter, so that x=t2 and y=2t.Then, the differential element of arc length ds is given by:ds=dx2+dy2=(2tdt)2+(2dt)2=2t2+1&amp;nbsp;dtSubstituting for x, y, and ds in the integral, we get:∫Cxy&amp;nbsp;ds=∫05(t2)(2t)(2t2+1&amp;nbsp;dt)Simplifying the integrand, we get:∫054t3t2+1&amp;nbsp;dtTo evaluate this integral, we can make the substitution u=t2+1, so that du=2t&amp;nbsp;dt and t2=u−1. Then, the integral becomes:∫1262(u−1)u&amp;nbsp;du=∫1262u3/2−2u1/2&amp;nbsp;duEvaluating the definite integral, we get:∫Cxy&amp;nbsp;ds=[45u5/2−43u3/2]126=3215(265/2−1)−323(263/2−1)Therefore, the value of the line integral is:∫Cxy&amp;nbsp;ds=3215(265/2−1)−323(263/2−1)

Evaluate the line integral, where C is the given curve C xy ds C: x=t^2 , y=2t ,0 \leq t \leq 5

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