Question

Evaluate the line integral ∮C F · dr, where F = Ÿ¨2x, 3y, zŸ© and C is given by the vector function r(t) = Ÿ¨t², t³, tŸ©.

Answer 1

To evaluate the line integral ∮C F · dr, the dot product of F and dr is calculated and then integrated over the given interval.

Step-by-step explanation:

To evaluate the line integral ∮C F · dr, where F = Ÿ¨2x, 3y, zŸ© and C is given by the vector function r(t) = Ÿ¨t², t³, tŸ©, we can follow these steps:

1. Calculate the dot product of F and dr.
2. Find the parameterization of C by substituting r(t) into the dot product formula.
3. Differentiate r(t) with respect to t to get dr/dt.
4. Substitute dr/dt into the dot product formula to get F · (dr/dt).
5. Integrate F · (dr/dt) with respect to t over the given interval.

In this case, F · dr evaluates to 15t⁴, so the line integral is given by the integral of 15t⁴ with respect to t over the interval of t from 0 to 1.