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What is the Formal Definition of the Definite Integral of the function #y=f(x)# over the interval #[a,b]# ?

Answer 1

Christopher Agresta

#int_a^bf(x)dx=lim_{n to infty}sum_{i=1}^n f(a+iDelta x)Deltax#, where #Delta x ={b-a}/n#

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Answer 2

Adam Adkins

The definite integral of the function ( y = f(x) ) over the interval ([a, b]) is defined as the limit of the Riemann sums as the width of the partitions approaches zero. Mathematically, it is denoted as:

[ \int_{a}^{b} f(x) , dx = \lim_{\Delta x \to 0} \sum_{i=1}^{n} f(x_i^*) \Delta x ]

where:

( \Delta x = \frac{b - a}{n} ) is the width of each subinterval,
( x_i^* ) is any point in the ( i )th subinterval ([x_{i-1}, x_i]).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.