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

a) y = -3x +2

b) y = 1/2x -6

these 2 can be combined into one equation.

-3x +2 = .5x -6

8 = 3.5x

x = 8 / 3.5 = 2.2857142857

y = -3 * (8 / 3.5) +2

y = (-24 / 3.5) + 2

y = -4.8571428571

step-by-step explanation:

answer: 80,000

step-by-step explanation:

the population doubles every day.

10,000*2=20,000 day 4

20,000*2=40,000 day 5

40,000*2=80,000 day 6