1. Define the equation of the circle and ellipse in the figure. 2.Find the area of the ellipse using integration.

Question

Using the given figure answer the following

1. Define the equation of the circle and ellipse in the figure. 
2. Find the area of the ellipse using integration. 
3. Find the area of the shaded region. (Use formula to find the area of the circle.

Answer 1

1. From the figure equation of the circle is x² + y² = 4 and that of the ellipse is \(\frac{x^2}{4} +\frac{y^2}{1} =1\).

2. We have,\(\frac{x^2}{4}+\frac{y^2}{1}=1\)

⇒  y² =1 ⇒ y =\(\frac{1}{2}\)\(\sqrt{4-x^2}\)

Area of the elipse = 4\(\int_0^2ydx\)

3. Area of the circle of radius 2 = π (2)² = 4π

∴ Area of the shaded region = Area of the circle – Area of the ellipse

= 4π – 2π = 2π.