Home page for accesible maths Math 101 Chapter 5: Integration

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5.5 Area as a limit

It can be shown that, when hh is very small, the lower sum arising from the lower rectangles and the upper sum arising from the upper rectangles are close together. One can define the integral of ff from aa to bb to be their common limit

∫abf⁢(x)⁢d⁢x=limh→0+⁡∑j=1nh⁢mj=limh→0+⁡∑j=1nh⁢Mj\int_{a}^{b}f(x)dx=\lim_{h\rightarrow 0+}\sum_{j=1}^{n}hm_{j}=\lim_{h% \rightarrow 0+}\sum_{j=1}^{n}hM_{j}

as the width of the strips goes to zero. This is called the Riemann integral. This process is used by computers to evaluate integrals numerically. See MATH114 for details