Finding the area of a circle using integration

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SUMMARY

The area of a circle can be calculated using integration techniques, specifically through trigonometric substitution or integration by parts. The equation of a circle is given by \((x-a)^2+(y-b)^2=r^2\), where \(r\) represents the radius. The area can be derived as \(2\int_{-r-a}^{r-a}y_{(x)}dx\), which represents the area of two half-circles. While the integration process may seem complex, it is a valid method for determining the area when the radius is unknown.

PREREQUISITES
  • Understanding of integration techniques, specifically trigonometric substitution
  • Familiarity with integration by parts
  • Knowledge of the equation of a circle
  • Basic calculus concepts, including definite integrals
NEXT STEPS
  • Study trigonometric substitution in integral calculus
  • Explore integration by parts with practical examples
  • Review the derivation of the area of a circle using calculus
  • Practice solving integrals involving circular equations
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and integration techniques, as well as anyone interested in geometric applications of integration.

satxer
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Let's say I have the equation for a circle but don't know how to calculate its radius. How could I use integration to find its area?
 
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satxer said:
Let's say I have the equation for a circle but don't know how to calculate its radius. How could I use integration to find its area?

You can always get the radius from circle equations !

However, all circle equations are integrated by trigonometric substitution
and it can also be done by integration by parts but that is a bit tricky!
 
The area of a circle is a equal to the area of two half-circle.

The equation of a half-circle (assuming (y-b) is always positive and that you don't understand that r is the radius):

\left(x-a\right)^2+\left(y-b\right)^2=r^2

So the area of a circle is equal to: 2\int_{-r-a}^{r-a}y_{(x)}dx

That's a complicated way, but I guess it can be done.
 

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