Finding the value of pi using calculus.

Question

I am new to calculus and not have much knowledge of how it works, I thought about finding the value of $\pi$ using the idea of dividing a circle into multiple parts and finding its area and thus extracting $\pi$ from it, the formula I got was $$\lim_{d \to 0} 4d\sum_{n=1}^{\lfloor 1/d\rfloor} \sqrt{1-(nd)^2} $$

Is there any way to simplify this formula using calculus as the only way currently I can solve this is taking $d$ as small value and adding each part manually.

Thanks for answers in advance

Answer 1

$\pi$ can be found using calculus as the arc length of a semicircle of radius $1$ (this method is similar to the one you suggest...). Since the circle has equation $x^2+y^2=1$, the arc length formula gives $$\pi=\int_{-1}^1\frac{1}{\sqrt{1-x^2}} dx$$