Question

Suppose F⃗ (x,y)=⟨ex,ey⟩F→(x,y)=⟨ex,ey⟩ and CC is the portion of the ellipse centered at the origin from the point (0,1)(0,1) to the point (7,0)(7,0) centered at the origin oriented clockwise. (a) Find a vector parametric equation r⃗ (t)r→(t) for the portion of the ellipse described above for 0≤t≤π/20≤t≤π/2.

Answer 1

Probably the intended ellipse is the one with equation

$\dfrac{x^2}{49}+y^2=1$

We can parameterize $C$ as a piece of this curve by

$\vec r(t)=\langle7\sin t,\cos t\rangle$

with $0\le t\le\frac\pi2$. Then

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^{\pi/2}\langle e^{7\sin t},e^{\cos t}\rangle\cdot\langle7\cos t,-\sin t\rangle\,\mathrm dt$

etc