योगफल कि सीमा के रूप में ` int _(0)^(3) (x^(2) - 2x + 2) dx ` का मान ज्ञात करे ।

Explanation

` int _(a)^(b) f(x) dx = lim _(h to 0 ) h sum _(r=1)^(n) f(a+rh)" "` …(1)
जहाँ , `nh = b-a` तथा ` n to oo`
यहाँ `f(x) = x^(2) - 2x +2, a = 0 ` तथा b=3
` :. f(a+rh) = f(0+rh) = f(rh) = r^(2) h^(2) - 2rh +2 `
साथ ही ` nh = b-a = 3 -0 = 3`
अब (1 ) से , ` int _(0)^(3) (x^(2) - 2x +2) dx = lim_(hto 0 ) sum _(r=1)^(n) h (r^(2)h^(2) - 2rh +2)`
` = lim _(hto0) sum_(r=1)^(n) (h^(3)r^(2) - 2h^(2)r + 2h)`
` = lim _( h to 0) (h^(3) sum _(r=1)^(n) r^(2) - 2h^(2) sum_(r=1)^(n) r + 2h sum_(r=1)^(n)1)`
` = lim _(h to 0 ) [ h^(3) (1^(2) + 2^(2) + ....+ n^(2)) - 2h^(2) (1+2+3 +...+ n) +2hn]`
` = lim _(h to 0 ) [ h^(3) (n(n+1)(2n+1))/6 - 2h^(2) (n(n+1))/2 + 2nh ] `
` lim _(h to 0 ) [ (nh(nh+h)(2nh+h))/6 - nh (nh+h) +2nh] `
` = lim _(h to 0) [ (3(3+h)(2xx 3+h))/6 - 3 (3+h)+2xx3]" " [ :' nh =3] `
` = (3 xx 3 xx 6 )/6 - 3 xx 3 + 2 xx 3 = 6 `