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.

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    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 :

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.

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

For more information, refer to the link given below:

link

WIth the parameterization

we end up with

We also have

So in the line integral, we get

step-by-step explanation:

the shape,the size and the distance