3. Let $u = x^2 + 4y^2$ define a potential. Calculate the line integral of $ abla u$ from $P_1(1,1)$ to $P_2(2,2)$ (a) by integrating along straight lines from $(1,1)$ to $(2,1)$ to $(2,2)$ and (b) by integrating along the straight line joining $P_1$ and $P_2$.
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Segment 1: From (1,1) to (2,1) We can parameterize this segment as r(t) = (1+t, 1), where 0 ≤ t ≤ 1. The differential of r(t) is dr(t) = (dt, 0). Substituting u = x + 4y into r(t), we get u = (1+t) + 4(1) = 5 + t. The line integral along this segment is given by Show more…
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