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.

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WIth the parameterization

we end up with

We also have

So in the line integral, we get