Question

Evaluate the line integral

∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 ,

where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦^2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1]. 

Answer 1

parameter for of C,

x=t, y=e^t ,z=e^t .

r(t)=t i+ e^t j+e^t k

=> dr=(i + e^t j + e^t k)dt

Then,

$\int_0^1 u.dr\\ =\int_0^1 (e^{2t},t,e^{t}).(1,e^t,e^t)dt\\ =\int_0^1 (2e^{2t}+te^{t})dt\\ =[ e^{2t}+te^{t}-e^t]_0^1\\ =e^2+e-e-1-0+1\\ =e^2$