Evaluate the following integrals as limit of sum: `int_3^5(2-x)dx`

Explanation

`I=int_3^5(2-x)dx`
`f(x)=2-x`
`int_3^5(2-x)dx=int_a^b f(x) dx`
where `a=3` and `b=5`
`int_a^b f(x) dx`
`=lim_(n->oo)[sum_(r=1)^n h *f(a+rh)]`
`f(a+rh)=f(3+rh)`
`=>f(3+rh)=2-3+rh`
`=-1+rh`
`int_3^5(2-x)dx=lim_(n->oo)[sum_(r=1)^n h *f(a+rh)]`
`=lim_(n->oo)[sum_(r=1)^n h*f(3+rh)]`
`=lim_(n->oo)[sum_(r=1)^n h(-1+rh)]`
`=lim_(n->oo) h(-n)+(n(n+1))/2 h^2`
`=lim_(n->oo)-(nh)+h^2n^2(1+1/n)/2`
`h=(b-a)/n`
`=>h=2/n`
`=>nh=2`
`=lim_(n->oo)(-2)+2^2(1+1/n)/2`
`=-2+4/2`
`=-2+2`
`=0`