Introduction to integrate area of circle:
A circle is a locus of a point moving at a constant distance from a fixed point. The fixed point is called as center of the circle and the constant distance is called the radius of the circle. When the fixed point moves a complete rotation, the surface covered by it is the area of the circle. If you divide the circle into a number of strips, integration of all the areas of the strips gives the area of the circle. In a rectangular coordinate system a circle is described with center at origin.
Description of Integrate Area of Circle
In a Cartesian system a circle normally symmetrical over both the rectangular axes. It means the center is at origin. If r is the radius of the circle, the equation of the circle is given by,
x2 + y2 = r2
The following graph describes the equation of a circle.
description of integrate area of circle
Therefore, at any point on the graph, the value of y is given by,
y = ±$ \sqrt{r^2 - x^2}$
Since we are considering the first quadrant, both the values of x and y are positive. Hence let us consider only the positive value of y.Understanding Calculus problem solver is always challenging for me but thanks to all math help websites to help me out.
Area of Circle by Integration
Area of circle by integration
Consider a small strip of a rectangle in one quadrant of the circle as shown above. Let the height of the rectangle be y and its tiny width be dx. Hence area of the rectangle is,
dA = (y)(dx).
If you integrate the above, you get the area of this quadrant of the circle. That is,
A = $\int_{0}^{r}(r^2 - x^2)^\frac{1}{2}$
=[(x/2) $ \sqrt{r^2 - x^2}$ + (r2/2)sin-1(x/r)]r0
= [(r/2)(0) + (r2/2)(Π/2)] – [0]
= (Π r2/4)
Since the circle is symmetrical, the area of this quadrant of the circle is four times the above area.
Therefore, the total area of the circle is, 4(Π r2/4) = Π r2
A circle is a locus of a point moving at a constant distance from a fixed point. The fixed point is called as center of the circle and the constant distance is called the radius of the circle. When the fixed point moves a complete rotation, the surface covered by it is the area of the circle. If you divide the circle into a number of strips, integration of all the areas of the strips gives the area of the circle. In a rectangular coordinate system a circle is described with center at origin.
Description of Integrate Area of Circle
In a Cartesian system a circle normally symmetrical over both the rectangular axes. It means the center is at origin. If r is the radius of the circle, the equation of the circle is given by,
x2 + y2 = r2
The following graph describes the equation of a circle.
description of integrate area of circle
Therefore, at any point on the graph, the value of y is given by,
y = ±$ \sqrt{r^2 - x^2}$
Since we are considering the first quadrant, both the values of x and y are positive. Hence let us consider only the positive value of y.Understanding Calculus problem solver is always challenging for me but thanks to all math help websites to help me out.
Area of Circle by Integration
Area of circle by integration
Consider a small strip of a rectangle in one quadrant of the circle as shown above. Let the height of the rectangle be y and its tiny width be dx. Hence area of the rectangle is,
dA = (y)(dx).
If you integrate the above, you get the area of this quadrant of the circle. That is,
A = $\int_{0}^{r}(r^2 - x^2)^\frac{1}{2}$
=[(x/2) $ \sqrt{r^2 - x^2}$ + (r2/2)sin-1(x/r)]r0
= [(r/2)(0) + (r2/2)(Π/2)] – [0]
= (Π r2/4)
Since the circle is symmetrical, the area of this quadrant of the circle is four times the above area.
Therefore, the total area of the circle is, 4(Π r2/4) = Π r2
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