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Engineering Mathematics

Line Integrals I

The line inte...

Question

The line integral of the vector field F = 5xz $^i + (3 x^{2} + 2 y)^j + x^{2} z^k$ along a path from (0,0,0) to (1,1,1) parametrized by $(t, t^{2}, t)$ is _______
4.17

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Solution

The correct option is A 4.17
Given vector is
$F = 5 x z^i + (3 x^{2} + 2 y)^j + x^{2} z^k$
$F . d l = 5 x z d x + (3 x^{2} + 2 y) d y + x^{2} z d z$
$∴ d x = d t, d y = 2 t d t, d z = d t$
At (0,0,0), t = 0 and at (1,1,1). t = 1
Hence the integral is
$\Rightarrow \int F . d l = \int_{0}^{1} [5. t . t . d t + (3 (t^{2}) + 2 t^{2}) 2 t d t + t^{2} t d t]$
$= \int_{0}^{1} (5 t^{2} + 5 t^{2} .2 t + t^{3}) d t$
$= \int_{0}^{1} (5 t^{2} + 11 t^{3}) d t$
$= {[\frac{5}{3} t^{3} + \frac{11}{4} t^{4}]}_{0}^{3}$
$= \frac{5}{3} + \frac{11}{4} = 4.167 \approx 4.17$

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