Evaluate ₂^3 x^2 d x as the limit of a sum

Step-by-step solution

aligned & Let I= ₂^3 x^2 d x & I= ₂^3 x^2 d x= ₂^3 f(x) d x , where f(x)=x^2 & Also, a=2, b=3 & _ h 0 h[f(2)+f(2+h)+f(2+2 h) & where, n h=b-a=3-2=1 & = _ h 0 h [2^2+(2+h)^2+(2+2 h)^2+ +(2+(n-1) h)^2 ] & = _ h 0 h [2^2+ (2^2+h^2+4 h )+ (2^2+2^2 h^2+8 h ) . & = _ h 0 h [2^2 h+h^2 (1^2+2^2+ . .+(n-1)^2 ) . & .= _ h 0 [4 n h+ h^3(n-1) n(2(n-1)+2+1) 6 +4+(n-1) ) ] & = _ h 0 [4 n h+ (n h-h)(n h)(2 n h-h) 6 +2(n h-h)(n h) ] & = _ h 0 [4 1+ (1-h) 1(2-h) 6 +2(1-h) 1 ] & =4+ 2 6 +2=6+ 1 3 = 19 3 aligned