Question

Evaluate the line integral ∫cf⋅dr, where f(x,y,z)=5xi−4yj+3zk and c is given by the vector function r(t)=⟨sint,cost,t⟩, 0≤t≤3π/2.

Answer 1

WIth the parameterization

$\mathbf r(t)=\langle\sin t,\cos t,t\rangle$

we end up with

$\mathbf f(\mathbf r(t))=\mathbf f(\sin t,\cos t,t)=5\sin t\,\mathbf i-4\cos t\,\mathbf j+3t\,\mathbf k$

We also have

$\mathrm d\mathbf r=\mathbf r'(t)\,\mathrm dt=\langle\cos t,-\sin t,1\rangle\,\mathrm dt$

So in the line integral, we get

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=3\pi/2}\mathbf f(\mathbf r(t))\cdot\mathbf r'(t)\,\mathrm dt$

$=\displaystyle\int_{t=0}^{t=3\pi/2}(9\sin t\cos t+3t)\,\mathrm dt=\frac98(3\pi^2+4)$

Answer 2

The line integral ∫cf⋅dr, where f(x,y,z)=5xi−4yj+3zk and c is given by the vector function r(t)=⟨sint , cost , t⟩, 0≤t≤3π/2 is  $\rm \dfrac{9}{8}(3\pi^2+4)$  and this can be determined by integrating and simplifying by substituting the limits.

Given :

f(x,y,z)=5xi−4yj+3zk and c is given by the vector function r(t)=⟨sint,cost,t⟩, 0≤t≤3π/2.

The following calculation can be used in order to evaluate the line integral ∫cf⋅dr :

$\rm f(r(t)) = f(sint,cost,t)$

$\rm f(sint,cost,t) = 5sint\; i - 4cost\; j +3t\;k$

Now, the expression of dr in terms of dt is given by:

dr = r'(t) dt

Now, substitute the known values in the line integral and simplify it.

$\rm \int\limits_c {x} \, dx = \int\limits^{t = 3\pi/2}_{t = 0} {f(r(t)). r'(t)} \, dt$

$\rm \int\limits_c {x} \, dx = \int\limits^{t = 3\pi/2}_{t = 0} {(9sintcost+3t)} \, dt$

Now, integrate both sides in the above expression and then simplify that expression by putting the limits.

$\rm \int\limits_c {x} \, dx = \dfrac{9}{8}(3\pi^2+4)$

For more information, refer to the link given below:

https://brainly.com/question/22008756