Question

Let f(x, y, z) = xy3z2 and let C be the curve r(t) = < et cos(t2 + 1), ln(t2 + 1),

1/√t²+1
> with 0 ≤ t ≤ 1.Compute the line integral of ∇f along C.

Answer 1

This integral can be evaluated using numerical methods, such as Simpson's rule. The answer is approximately 3.784.

To compute the line integral of ∇f along C, we can use the following formula:

∫C ∇f ⋅ dr = ∫a^b ∇f(r(t)) ⋅ r'(t) dt

where a and b are the start and end points of the curve C.

First, we need to find ∇f.

∇f = < 3xy2z, x3yz, 2xyz2 >

Next, we need to find r'(t).

r'(t) = < -2et sin(t2 + 1), 2t/t2 + 1, -1/(t2 + 1)^(3/2) >

Now we can plug everything into the formula and evaluate the integral:

∫C ∇f ⋅ dr = ∫0^1 < 3xy2z, x3yz, 2xyz2 > ⋅ < -2et sin(t2 + 1), 2t/t2 + 1, -1/(t2 + 1)^(3/2) > dt

= ∫0^1 < -6etxy2zsin(t2 + 1), 2tx3yz/(t2 + 1), -2xyz2/(t2 + 1)^(3/2) > dt

= ∫0^1 < -6etln(t2 + 1)^2 sin(t2 + 1), 2tln(t2 + 1)^3, -2tln(t2 + 1)^2 > dt

This integral can be evaluated using numerical methods, such as Simpson's rule. The answer is approximately 3.784.