Question 2. Applications of Trig Substitution: An Ellipse and the Solar Eclipse (a) Show that the area of the region bounded by the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is given by $\pi a b$. Deduce the formula for the area bounded inside a circle of radius $r$ as a simple consequence. 5

Step 1 ：Given the equation of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), we want to find the area of the region bounded by this ellipse.

Step 2 ：The area of an ellipse is given by the formula \(\pi a b\), where \(a\) and \(b\) are the semi-major and semi-minor axes of the ellipse respectively.

Step 3 ：For a circle, the semi-major and semi-minor axes are the same and equal to the radius \(r\) of the circle.

Step 4 ：Therefore, the area of a circle can be deduced from the formula for the area of an ellipse by substituting \(a = b = r\).

Step 5 ：\(\text{area}_\text{ellipse} = \pi a b\)

Step 6 ：\(\text{area}_\text{circle} = \pi r^{2}\)

Step 7 ：\(\boxed{\text{Final Answer: The area of the region bounded by the ellipse }\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\text{ is given by }\pi a b\text{. The area of a circle of radius }r\text{ can be deduced from this formula by substituting }a = b = r\text{, which gives the area as }\pi r^{2}\text{.}}\)