Evaluate: `int_a^bsinx dx` using limit of sum

Explanation

We have
`int_(a)^(b)=sin x dx=limh[sina+sin(a+h)+sin(a+2h)+……..`
`+sin{a+(n-1)h}]` where `nh=b-a`
`=lim_(nto oo) [(sin{a+1/2(n-1)h}"sin"(1/2nh))/("sin"(1/2h))]`
`=lim_(hto 0) (2sin{a+1/2(nh-h)}"sin"(1/2nh).(1/2h))/("sin"(1/2h))`
`=2sin{a+1/2(b-a-0)}."sin"1/2(b-a).1`
`[ :' "as" nto oo, h to 0, "and" nh=b-a]`
`=2sin {1/2(b+a)}"sin"{1/2(b-a)}`
`=cos a-cosb`