Question

Evaluate the line integral ∫CF⋅d r where F=⟨5sinx,−4cosy,10xz⟩ and C is the path given by r(t)=(t^3,−3t^2,2^t) for 0≤t≤1

∫CF⋅d r=

Answer 1

Answer: To evaluate the line integral ∫CF⋅dr, we first need to parameterize the path C in terms of a single variable, say t. The parameterization of C is given by:

r(t) = (t^3, -3t^2, 2^t), 0 ≤ t ≤ 1

We can then write the differential of the path as:

dr = (3t^2, -6t, 2^t ln(2)) dt

Next, we evaluate the dot product F ⋅ dr along the path C:

F(r(t)) ⋅ dr = ⟨5sin(t^3), -4cos(-3t^2), 10t^3(2^t)⟩ ⋅ ⟨3t^2, -6t, 2^t ln(2)⟩ dt

= 15t^2 sin(t^3) + 24t cos(3t^2) + 20t^3 (2^t) ln(2) dt

Finally, we integrate this expression over the interval 0 ≤ t ≤ 1 to get the value of the line integral:

∫CF ⋅ dr = ∫0^1 (15t^2 sin(t^3) + 24t cos(3t^2) + 20t^3 (2^t) ln(2)) dt

This integral cannot be evaluated exactly, but we can use numerical integration to approximate the value. Using a numerical integration tool, the value of the line integral is approximately equal to 11.108.

Therefore:

∫CF ⋅ dr ≈ 11.108

Step-by-step explanation:

Answer 2

To evaluate the line integral along a given path, perform dot product of the field with dr and integrate with respect to t from 0 to 1 after substituting the path's parametric equations into the vector field.

Step-by-step explanation:

The student has asked to evaluate the line integral of a vector field along a given path. To do this, we need to substitute the parametric equations of the path into the vector field, compute the derivatives of the parametric equations which will give us d r, and then perform the dot product of the field with d r. After that, we need to integrate the resulting expressions with respect to the parameter t from 0 to 1.

First, we find the components of F along the path C. Then integrate each component with its corresponding d r component. The integral can be expressed as ∫CF·d r = ∫C(Fxd x + Fyd y + Fzd z).

Finally, we evaluate this integral from t = 0 to t = 1 to find the work done by the force field along the path. The actual computation will involve substituting the parametric equations into F and integrating the resulting expression with respect to t.