Evaluate the following integrals using limit of sum.
`int_(a)^(b)cos x dx`

Answer

`sinb-sina`

Explanation

Here `f(x)=cosx`
`int_(a)^(b)cosx dx=lim_(n to oo) h [cosa +cos(a+h)+………..`
`+cos {a_(n-1)h}],` where `nh=b-a`
`=lim_(nto oo) h(cos{a+1/2(n-1)h}"sin"(1/2nh))/("sin"(1/2h))`
`=lim_(nto oo) [2cos{a+(1/2nh-1/2h)}sin(1/2nh)((1/2h)/(sin(1/2h)))]`
`=2cos {a+1/2(b-a)-0}.sin(1/2(b-a))xx1`
`=2"cos"1/2(a+b)"sin"1/2(b-a)`
`=sinb-sina`