why pi r2 is the area of a circle
Area of a circle
A r e a ( r ) = ∫ 0 r 2 π t d t = 2 π ( t 2 2 ) 0 r = π r 2 . ( \ displaystyle ( \ begin ( aligned ) \ mathrm ( Area ) ( r ) & ( ) = \ int _ ( 0 ) ^ ( r ) 2 \ pi t \ , dt \ \ & ( ) = 2 \ pi \ left ( ( \ frac ( t ^ ( 2 ) ) ( 2 ) ) \ right ) _ ( 0 ) ^ ( r ) \ \ & ( ) = \ pi r ^ ( 2 ). \ end ( aligned ) ) )
A r e a ( r ) = ∫ 0 r 2 π t d t = 2 π ( t 2 2 ) 0 r = π r 2 . ( \ displaystyle ( \ begin ( aligned ) \ mathrm ( Area ) ( r ) & ( ) = \ int _ ( 0 ) ^ ( r ) 2 \ pi t \ , dt \ \ & ( ) = 2 \ pi \ left ( ( \ frac ( t ^ ( 2 ) ) ( 2 ) ) \ right ) _ ( 0 ) ^ ( r ) \ \ & ( ) = \ pi r ^ ( 2 ). \ end ( aligned ) ) )