Question

Evaluate the line integral ∫_C F⋅dr, where F(x,y,z)=−4sin(x)i+3cos(y)j−4xzk and C is given by the vector function r(t)=t^6i−t^5j+t^4k , 0≤t≤1.

Answer 1

we are given

we can find dr

now, we can set up integral

now, we can solve it

now, we can solve each integrals and combine them

we can use u-subs

now, we can solve second integral

we can use u-subs

u=t^5

du=5t^4dt

we can plug back u

now, we can plug bounds

now, we can solve third integral

now, we can combine them

now, we can simplify it

and we get

...........Answer