Question

Find a vector parametric equation rƒ— (t) for the circle c that starts at the point (7,0) and travels around the circle once counterclockwise for 0≤t≤2π. Using your parametrization in part (a), set up an integral for calculating the circulation of fƒ— around c?

Answer 1

The vector parametric equation for the circle is r(t) = 7(cos 2πtĨ + sin 2πtĵ). The integral for calculating the circulation of f around the circle is ∫[f dot r'(t)] dt.

Step-by-step explanation:

To find the vector parametric equation for the circle, we can use the formula:

rƒ(t) = A(cos ωtî + sin ωtï)

Where A is the radius of the circle and ω is a constant representing the angular velocity. In this case, the point (7,0) corresponds to A = 7 and we're given that the circle travels around once counterclockwise, so ω = 2π.

Substituting these values into the equation gives:

rƒ(t) = 7(cos 2πtî + sin 2πtï)

To set up an integral for calculating the circulation of fƒ around the circle, we need to evaluate fƒ dot rƒ′(t) and integrate it over the path of the circle. The circulation is given by the integral:

∫[fƒ dot rƒ′(t)] dt