Evaluate ∫√(r^2-x^2)dx, x∈[0,r] where r is a fixed positive number. Hence prove the area of the circle of radius r

Question

Evaluate \(\int\limits_0^r\sqrt{r^2-x^2}dx\), where r is a fixed positive number. Hence prove the area of the circle of radius r is πr².
Find the area of the circle, x² + y² = 16 which is exterior to parabola y² = 6x.

Answer 1

1.

2. Given, x² + y² = 16 and y² = 6x ⇒ x² + 6x = 16 ⇒ x² + 6x – 16 = 0 ⇒ (x + 8)(x – 2) = 0 ⇒ x = -8, 2.

Area = Area of the circle – Interior area of the parabola.