I Integrating Sin(x): Solve the Area of a Half Circle

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The integration of sin(x) from 0 to π equals 2, which is a common calculus result. However, the discussion clarifies that interpreting this integral as the area of a half circle of radius one leads to confusion. The error lies in misidentifying the variable x as a coordinate rather than the angle subtended. The correct approach involves integrating (sin(x))^2, which accurately represents the area of the half circle, yielding π/2. This highlights the importance of understanding the geometric interpretation of integrals in calculus.
Karim Habashy
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Hi All,

This is a very elementary question but, from calculus :

∫sinx dx from 0 to pi = 2, but i thought of it, in terms of the attached driagram.
upload_2016-3-5_19-42-31.png


And if we think of it, this way, the integration is the area of a half circle of radius one and is equal to Pi/2.

Where did i go wrong ?
 
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Karim Habashy said:
And if we think of it, this way, the integration is the area of a half circle of radius one and is equal to Pi/2.
No, it's not. Your mistake was to assume ##x## as the coordinate in the horizontal line, when it's actually the angle subtended between the line and the horizontal axis.
 
You are right, thanks. the integration would be ∫(sinx)2 dx which is Pi/2.
 
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