Find the value of $\int_C F.dr$ for this ellipse

Question

Let $F(x,y)=\langle -y ,x\rangle$ and $C$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9} = 1 $ oriented counter clockwise, then find the value of $\int_C F.dr$

This is how I tried it,

$x=4\cos(t) \rightarrow dx=-4\sin(t)$

$y=3\sin(t) \rightarrow dy=3\cos(t)$

and $0\leq t\leq 2\pi$

$\int_C F.dr= \int-ydx +xdy = \int_{t=0}^{2\pi}12(\sin^2(t)+\cos^2(t))dt = 75.4$

But answer given to me at the back is $98.2$ and doesn't agrees with mine. Can someone tell me where I made the mistake? Or can anyone verify if my calculation is right?

Answer 1

Your solution is correct. In fact, $$ \int_C F(r) \cdot dr = \int_0^{2 \pi} (-3 \sin t, 4 \cos t)\cdot (-4 \sin t, 3 \cos t) dt = \int_0^{2 \pi} 12(\sin^2 t+ \cos^2 t) dt = 24 \pi \approx 75.3982 $$

There is a typo/mistake in the book. It happens sometimes.