Question

Heya mates!!!!!

Are u busy????

If not plzzz answer this question:


Prove that the circumference of a circle is 2πr.

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Answer 1

Hey Mate,

We know that $\pi$ is equal to the circumference of a circle divided by its diameter. Then, let us frame an equation.

=> $\pi$ = Circumference / Diameter

=> $\pi$ = Circumference / 2 ( Radius )

( Since, Diameter = 2 * Radius

Now,

On transposing Circumference to LHS and $\pi$ to RHS, we get :-

=> Circumference = $\pi2r$ = $2\pi r$

Hence, proved that Circumference of a Circle = $2\pi r$

Hope My Answer Brings A Smile On Your Face !

^_^

Answer 2

Hey mate!!!
Here's your answer...

If we imagine the circle centered in the origin with radius r, it has the equation:

x2+y2=r2, (in the graph the radius is 2):

graph{x^2+y^2=4 [-10, 10, -5, 5]}

or y=±√r2−x2

And considering the fourth of circle in the first quadrant, we can obtain the lenght of a line with the integral:

L=4∫r0√1+(y')2dx.

This integral is quite long, so we can parametrize the circle as usual:

x=rcosθ
y=rsinθ

and use this integral:

L=∫ba√[x'(θ)]2+[y'(θ)]2dθ.

Since:

x'=−rsinθ
y'=rcosθ

So:

L=4∫π20√r2sin2θ+r2cos2θdθ=

=4∫π20√r2(sin2θ+cos2θ)dθ=

=4∫π20rdθ=4[rθ]π20=4rπ2=2πr.

Hope this will help you.
Plz mark it as Brainliest...