# How do can you derive the equation for a circle's circumference using integration?

Let assume that the circle is centered at the origin hence its equation is

Using parametrization in two variables: we can write the same circle above as

and thus the arclength is given by (integration)

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To derive the equation for a circle's circumference using integration, we start by considering a quarter of a circle in the first quadrant. Let ( y = \sqrt{r^2 - x^2} ) be the equation of the upper half of the circle, where ( r ) is the radius.

We integrate this function from ( x = 0 ) to ( x = r ) to find the arc length of this quarter circle. The arc length ( S ) is given by the integral:

[ S = \int_{0}^{r} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx ]

[ = \int_{0}^{r} \sqrt{1 + \left(\frac{d}{dx}\sqrt{r^2 - x^2}\right)^2} dx ]

[ = \int_{0}^{r} \sqrt{1 + \left(\frac{-x}{\sqrt{r^2 - x^2}}\right)^2} dx ]

[ = \int_{0}^{r} \sqrt{1 + \frac{x^2}{r^2 - x^2}} dx ]

[ = \int_{0}^{r} \sqrt{\frac{r^2 - x^2 + x^2}{r^2 - x^2}} dx ]

[ = \int_{0}^{r} \sqrt{\frac{r^2}{r^2 - x^2}} dx ]

Now, let ( x = r\sin(\theta) ), then ( dx = r\cos(\theta)d\theta ), and the integral becomes:

[ S = \int_{0}^{\frac{\pi}{2}} \sqrt{\frac{r^2}{r^2 - r^2\sin^2(\theta)}} r\cos(\theta)d\theta ]

[ = \int_{0}^{\frac{\pi}{2}} \sqrt{\frac{r^2}{r^2(1 - \sin^2(\theta))}} r\cos(\theta)d\theta ]

[ = \int_{0}^{\frac{\pi}{2}} \sqrt{\frac{r^2}{r^2\cos^2(\theta)}} r\cos(\theta)d\theta ]

[ = \int_{0}^{\frac{\pi}{2}} \frac{r}{|\cos(\theta)|} r\cos(\theta)d\theta ]

[ = r \int_{0}^{\frac{\pi}{2}} d\theta ]

[ = r\left[\theta\right]_{0}^{\frac{\pi}{2}} ]

[ = r\left(\frac{\pi}{2} - 0\right) ]

[ = \frac{\pi r}{2} ]

Since the quarter circle has symmetry, the circumference ( C ) of the full circle is four times this value:

[ C = 4 \times \frac{\pi r}{2} = 2\pi r ]

Thus, we have derived the equation for the circumference of a circle using integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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