what is the area of circle using integration method
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what is the area of circle using integration method
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The equation of a circle with radius, r, is given by
.
Making
the subject of the formula, we have
.
Now, the radius of a circle divides the circle into four equal parts called the quadrants. And the area of the portion bounded by a curve between two points on the x-axis is given by the integral between the two points of the curve.
Thus, the area of a quadrant of a circle bounded by the center of the circle and the circumference (that is, the radius) is given by
[Note that Area is always positive so we take the positive value of
.]
Thus, the area of the 4 quadrants of a circle is given by
.
Converting to polar coordinates,
. When
,
and when
, 
Thus,
Using the two trigonometric identities
and
, we have that
.
Thus we have,
Therefore, the area of a circle with a radius of r is given by
.