Evaluate the following definite integrals as limit of sum: `int_0^4x\ dx`

Explanation

We know that,

`int_{a}^{b} f(x) d x=(b-a) lim _{n rightarrow infty} frac{1}{n}[f(a)+f(a+h)+ldots+f(a+(n-1) h)],`
where `h=frac{b-a}{n}`


Here we have , `a=0, b=4,` and `f(x)=x`

So, `h=frac{4-0}{n}=frac{4}{n}`

`rArr int_{0}^{4}(x) d x=(4-0) lim _{n rightarrow infty} frac{1}{n}} frac{1}{n}[f(0)+f(h)+f(2 h)+ldots+f((n-1) h)]`

`=4 lim _{n rightarrow infty} frac{1}{n}[(0)+(h)+(2 h)+ldots+{(n-1) h}]`

`=4 lim _{n rightarrow infty} frac{1}{n}[1+(h)+(2 h)+ldots+{(n-1) h]`

`=4 lim _{n rightarrow infty} frac{1}{n}[{h+2 h+3 h+ldots+(n-1) h}]`

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