Use a line integral to compute the area of the region bounded by the ellipse .

Question

Use a line integral to compute the area of the region bounded by the ellipse  .

Quizplus · Accepted Answer

We can parameterize the ellipse as 
$$\vec{r}(t) = \langle a\cos(t), b\sin(t)angle,\quad 0\leq t \leq 2\pi$$
where $a$ and $b$ are the major and minor axes, respectively. In this case, $a=3$ and $b=2$, so 
$$\vec{r}(t) = \langle 3\cos(t), 2\sin(t) angle,\quad 0\leq t \leq 2\pi$$
The area of the ellipse can be found using a line integral by computing
$$A = \frac{1}{2}\oint_{C} xdy-ydx$$
where $C$ is the boundary of the ellipse, which is given by $\vec{r}(t)$. Evaluating this integral yields
\begin{align*}
A &= \frac{1}{2}\int_{0}^{2\pi} \left( 3\cos(t) \cdot 2\cos(t) - 2\sin(t)\cdot(-3\sin(t)) ight) dt \
&= \frac{1}{2}\int_{0}^{2\pi} 6\cos^2(t) + 6\sin^2(t) dt \
&= \frac{1}{2}\int_{0}^{2\pi} 6 dt \
&= 3\pi
\end{align*}
Therefore, the correct answer is choice C.

Use a Line Integral to Compute the Area of the Region