Evaluate the following definite integrals as limit of sums. `int_a^ b x""dx`

Explanation

Let `I=int_a^b x dx`
We know that
`lim_(h->0)[f(a)+f(a+h)+f(a+2h)+....+f(a+(n−1)h)]`, where `h=(b−a)/n`
Here, `f(x)=x`
`∴int_a^b=lim_(x->0)h[a+(a+h)+(a+2h)+...+(a+(n−1)h]`
`=>int_a^b x dx lim_(x->0)[na+{h+2h+3h+...+(n−1)h}]`
`=>int_a^b x dx lim_(x->0)[(nh)a+h^2(n(n−1))/2=lim_(x->0)[(nh)a+(nh)(nh−h)/2]]`
`=>int_a^b x dx lim_(x->0)[(b−a)a+((b−a)(b−a−h))/2]=(b−a)a+((b−a)^2)/2=(b^2−a^2)/2`