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

`\mathbf{ - (3 \pi^2)/(2)}`

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 [\frac {t^2}{2}]^(\pi)_(0) \ \ dt`

`= 10 [(sin^2 \ t)/(2)]^(\pi)_(0) - (3)/(2)(\pi)^2`

By dividing 2 with 10 and integrating
`= 10 [(sin^2 \ t)/(2)]^(\pi)_(0)`; we have:

`=5(sin^2t -sin^2 0) -(3 \pi^2)/(2)`

`=5(0) -(3 \pi^2)/(2)`

`= 0 - (3 \pi^2)/(2)`

`\mathbf{= - (3 \pi^2)/(2)}`