Finding the Circumference of a Circle Using Integrals
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Finding the Circumference of a Circle Using Integrals
This paper shows the relationship between calculus and circumference of a circle. There are two significant concepts linking the circumference of a circle and integration. First, it shows how the combination is used to derive the circumference formula 2 of a circle. Secondly, it shows proof that the circumference of a circle is equal to 2π using integration.
To achieve the first part, the equation of a circle and the arc length formula are applied. The exact length of an arc can be determined by integration using the formulae below. In this paper, it will be proved that it truly holds. The formula states that given a continuous function of and closed within a closed interval [a, b], then the length of an arc from a point to a point is obtained by the integral, to, where is the derivative of.
To compute the circumference of a circle using integrals, the limits are taken from to. They are the limits of a semi-circle.
Consider a circle with the radius r and a point (x, y) a point on its circumference, the x and y being values read from the X-axis and Y-axis, respectively. Applying the arc length formula and the radius, the following equation holds.
When y is made the subject of the formula, the equation becomes,
The arc length of half the circle is obtained by (Shenitzer & Steprāns, 1994).
But
Therefore,
=dx
=dx
=dx
=
The next step is to Plug in the values of r and –r.
But the value ofis equal to. Therefore,
=
=
=
=πr.
The value πr is the circumference of the half of the circle. It is multiplied by 2 to obtain the value of the whole circumference.
i.e. =2 πr.
That is the end of the proof of how to use integrals to derive the circumference formula.
The following part shows a proof, using integrals that the circumference of a unit circle equal to.
In a unit circle, any point (x, y) is such that x is defined as cos, and y is defined as sin. That is;
X= cos
Y=sin
In finding the circumference of a circle using integrals, the arc length formula is used. That is d.
It is necessary to start by getting the derivatives of x and y with respect to.
From X= cos
Equivalently,
They are the plugged in to the arc length formula and the limits of a=0 and b=.
It results in the integral below.
d.
From trigonometry, Pythagorean, the value ofis equal to.
Then,
=d.
The value 1 can be retained or ignored.
By integrating, the following result is obtained.
=
Substituting in the values of and 0, the final result is achieved.
=
=
Reference List
Shenitzer, A., & Steprāns, J. (1994). The evolution of integration. The American Mathematical Monthly, 101(1), 66-72.