what is the area of circle using integration method

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This answer was edited.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 Pythagorean trigonometric identity .]

Using the two trigonometric identities and , we have that .

Thus we have,

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Therefore, the area of a circle with a radius of r is given by .