Coming Up With The Formula For Areas Underneath Curves

Area\quad PMTN<\delta A<Area\quad SMTQ\\ \\ y\delta x<\delta A<\left( y+\delta y \right) \delta x\\ \\ y<\frac { \delta A }{ \delta x } <y+\delta y\\ \\ Now...\\ \\ \lim _{ \delta x\rightarrow 0 }{ \left( \frac { \delta A }{ \delta x }  \right) =\frac { dA }{ dx }  } \\ \\ \lim _{ \delta x\rightarrow 0 }{ \left( \frac { \delta A }{ \delta x }  \right) =y } \\ \\ because,\quad when\quad \delta x\rightarrow 0,\quad \delta y\rightarrow 0.\\ \\ so...\quad y=\frac { dA }{ dx } \\ \\ 1dA=ydx\\ \\ \int { 1dA=\int { ydx }  } \\ \\ A=\int { ydx } =F\left( x \right) +C\\ \\ But\quad at\quad x=a,\quad A=0,\\ \\ 0=F\left( a \right) +C\\ \\ \therefore \quad C=-F\left( a \right) \\ \\ So\quad A=\int _{ a }^{ x }{ ydx=F\left( x \right)  } -F\left( a \right) ,\\ \\ but\quad you\quad want\quad to\quad find\quad out\quad the\quad area\quad from\quad x=b\quad to\quad x=a,\\ so\quad you\quad use:\\ \\ A=\int _{ a }^{ b }{ ydx=F\left( b \right) -F\left( a \right)  } .

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