Line Integral (4-2-2020)

$\red{\vec{F} = \hat{i} + 2y \ \hat{j} + 3z^2 \ \hat{k}} \ ; \ \blue{d\vec{l} = dx \ \hat{i} + dy \ \hat{j} + dz \ \hat{k}}$ $\implies \vec{F} \cdot d\vec{l} = dx + 2y \ dy + 3z^2 \ dz = d\left(x + y^2 + z^3\right)$

The following integral is to be computed between points $(1,1,1)$ and $(2,3,4)$ along the line joining them. Since the dot product is an exact differential, the answer is path independent.

$\vec{F} \cdot d\vec{l} =\int_{\mathrm{initial}}^{\mathrm{final}} d\left(x + y^2 + z^3\right)$

Let: $x + y^2 + z^3 = t$ . Recomputing the limits and applying the substitution transforms the integral to:

$\boxed{\int_{3}^{75} dt = 72}$