## Fundamental Theorem of Calculus

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### Solution

The Riemann method includes approximating the area under a curve by rectangular elements.

Left-Riemann sum (Area of left-rectangles) :

Right-Riemann sum (Area of right-rectangles) :

Therefore :

\[\lim_{n \rightarrow \infty}\sum_{i=0}^{n} f\left(x_i\right) \Delta x=\int_a^b f(x) d x\]

Relationship between integration and differentiation :

\[
F(b)-F(a)=\sum_{i=1}^n\left[F\left(x_i\right)-F\left(x_{i-1}\right)\right]
\]

\[\int_a^b f(x) d x=F(b)-F(a)\]

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