Question

Evaluate R C F · dr, where F(x, y, z) = 5xi − 5yj − 3zk and C is given by the vector function r(t) = hsin t, cost, ti, where 0 ≤ t ≤ π.

Answer 1

Answer:

$\mathbf{ - \dfrac{3 \pi^2}{2}}$

Step-by-step explanation:

Given that:

F(x, y, z) = 5xi - 5yj - 3zk

The objective is to evaluate the $\int _c F \ dr .C$

and C is given by the vector function  r(t) = (sin t, cost, t) where  0 ≤ t ≤ π

$F(r(t)) = 5 \ sint \ i - 5 \ cost \ j - 3t \ k$

∴

$\int_c F . \ dr = \int ^{\pi}_{0} ( 5 \ sint \ i - 5 cos t \ j - 3 t \ k ) ( cos \ t , - sin \ t , 1 ) \ dt$

$=\int ^{\pi}_{0} ( 5 \ sint \ cost+ 5 cos t \ sin t - 3 t) dt$

$=\int ^{\pi}_{0} ( 10 \ sint \ cost) \ dt -3 \int ^{\pi}_{0} \ dt$

$= \int ^{\pi}_{0} ( 10 \ sint \ cost) \ dt - 3 [\dfrac {t^2}{2}]^{\pi}_{0} \ \ dt$

$= 10 [\dfrac{sin^2 \ t}{2}]^{\pi}_{0} - \dfrac{3}{2}(\pi)^2$

By dividing 2 with 10 and integrating $= 10 [\dfrac{sin^2 \ t}{2}]^{\pi}_{0}$; we have:

$=5(sin^2t -sin^2 0) -\dfrac{3 \pi^2}{2}$

$=5(0) -\dfrac{3 \pi^2}{2}$

$= 0 - \dfrac{3 \pi^2}{2}$

$\mathbf{= - \dfrac{3 \pi^2}{2}}$