Example 2.51. Calculate the area of the figure bounded by the ellipse $x=4 \cos t, y=3 \sin t$.
Solution. We will use the last equality from (2.22).

$$
\begin{gathered}
S=\frac{1}{2} \int_{0}^{2 \pi}\left[x(t) y^{\prime}(t)-y(t) x^{\prime}(t)\right] d t= \\
=\frac{1}{2} \int_{0}^{2 \pi}[4 \cos t \cdot 3 \cos t-3 \sin t(-4 \sin t)] d t=6 \int_{0}^{2 \pi} d t=12 \pi
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_0f84e17422d7c0df8d68g-081.jpg?height=296&width=474&top_left_y=876&top_left_x=125)

Fig. 2.6.