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

abondantQ

abondantQ

Answered question

2021-05-11

Evaluate the line integral, where C is the given curve
C xy ds
C: x=t2,y=2t,0t5

Answer & Explanation

Dora

Dora

Skilled2021-05-12Added 98 answers

Integral solution:

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Jeffrey Jordon

Jeffrey Jordon

Expert2023-04-30Added 2605 answers

To evaluate the line integral Cxyds, where C is the curve C:x=t2,y=2t,0t5, 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 0t5.
Next, we need to express ds in terms of dt. We have
ds=(dxdt)2+(dydt)2dt.
Substituting x=t2 and y=2t into this equation, we get
ds=(2t)2+(2)2dt=4t2+4dt=2t2+1dt.
Now we can evaluate the line integral:
Cxyds=05(t2)(2t)(2t2+1)dt=405t3t2+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=12126(u1)3/2du.
Using the power rule for integration, we can evaluate this integral to get
12126(u1)3/2du=12·25(u1)5/2|126=15(26677825).
Therefore, the value of the line integral is 15(26677825).
RizerMix

RizerMix

Expert2023-04-30Added 656 answers

The given line integral can be expressed as:
Cxy ds
where C is the curve defined by x=t2, y=2t, and 0t5.
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 dt
Substituting for x, y, and ds in the integral, we get:
Cxy ds=05(t2)(2t)(2t2+1 dt)
Simplifying the integrand, we get:
054t3t2+1 dt
To evaluate this integral, we can make the substitution u=t2+1, so that du=2t dt and t2=u1. Then, the integral becomes:
1262(u1)u du=1262u3/22u1/2 du
Evaluating the definite integral, we get:
Cxy ds=[45u5/243u3/2]126=3215(265/21)323(263/21)
Therefore, the value of the line integral is:
Cxy ds=3215(265/21)323(263/21)

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