Evaluate the following definite integrals as limit of sum: `int_0^2(2x^2+3)dx`

Explanation

let us assume ,

`f(x) =2 x^{2}+3 x`
`a=0, b=3, n h=3-0=3`
`therefore int_{0}^{3}(2 x^{2}+3 x) d x=lim _{h arrow 0} h[f(0)+f(0+h)+f(0+2 h)+cdots+f(0+(n-1) h)]`

` =lim _{h arrow 0} h[(2 h^{2}+3 h)+(8 h^{2}+6 h)+cdots+{2(n-1)^{2} h^{2}+3(n-1) h}`

` =lim _{h arrow 0} h[{3 h(1+2+3 cdots+(n-1))}+2 h^{2}(1^{2}+2^{2}+3^{2}+cdots+(n-1)^{2} h^{2})]`


` =lim _{h arrow 0} h[frac{3 h n(n-1)}{2}+frac{..h^{3} / 2 n^{3}-3 n^{2}+n)]}{3}]`

` =lim _{h arrow 0} h[frac{3(n h)^{2}}{2}-frac{3(n h) h}{2}+frac{2(n h)^{3}}{3}-frac{3(n h)^{2}}{3} cdot h+frac{(n h) cdot h^{2}}{3}]`


` =frac{3(3)^{2}}{2}-frac{3 cdot 3 cdot 0}{2}+frac{2(3)^{3}}{3}-frac{3(3)^{2} cdot 0}{3}+frac{3 cdot 0}{3}`


` =frac{27}{2}+18`


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