Parametric equation of curve C is r(t)=<t,2t2>,0≤t≤1
Here, differential length ds is,
ds=∥r′(t)∥dt
=∥<1,4t>∥dt
=12+(4t)2dt
=1+16t2dt
Now, the line integral is evaluated as,
∫Cf(x,y)ds=∫C8y+1ds
=∫018(2t2)+11+16t2dt
=∫011+16t21+16t2dt
=∫01(1+16t2)dt
=[t+16(3t3)]01
=1+316
=319
Hence, option (c) is correct.
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