Evaluate the following integrals as limit of sum: `int_0^2(x+4)dx`

Explanation

Given: `int_0^2(x+4)dx`

`I=int_a^b f(x)dx=(b-a)lim_(n->oo)(1/n)[f(a)+f(a+h)+......+f(a+(n-1)h)]`

Here, `a=0,b=2, f(x)=x+4, h=(b-a)/n=(2-0)/n=2/n`

`I=lim_(n->oo)(1/n)[f(0)+f(0+2/n)+......+f(0+(n-1)2/n)]`

`I=lim_(n->oo)(2/n)[0+4+2/n+4+2(2/n)+4+......+(n-1)2/n+4]`

`I=lim_(n->oo)[(2/n)(4n+(2+2(2)+3(2)......(n-1)2))]`

`I=lim_(n->oo)[(2/n)(4n+(2(1+2+.....+(n-1)))/n]`

`I=lim_(n->oo)[(2/n)[4n+2((n-1)n))/(2n)`

...