Question

Find the area of the circle \(x^2+y^2=r^2\).

Hint:

Integrate \(4\int_0^r\sqrt{r^2-x^2}\,dx\) using the standard formula, or sum concentric rings \(2\pi\rho\,d\rho\).

Answer 1

Step 1: Polar-style slicing (thin rings) instead:
Think of the disc as built from thin concentric rings of radius \(\rho\) (from 0 to \(r\)) and thickness \(d\rho\); each ring's area is approximately its circumference times thickness, \(2\pi\rho\,d\rho\).

Step 2: Integrating over all rings:
Total area \(=\displaystyle\int_0^r 2\pi\rho\,d\rho=2\pi\left[\dfrac{\rho^2}{2}\right]_0^r=2\pi\cdot\dfrac{r^2}{2}=\pi r^2\).

Final Answer:
Same result via ring-summation: \(\boxed{\pi r^2}\).