Question

If f(x)=x^2
Find
lim dx-->0
dx{ f(1)+f(1+dx) + f(1+2dx) + f(1+3dx)+...+f(2)}

Answer 1

Without using integration

Answer 2

dx{ f(1)+f(1+dx) + f(1+2dx) + f(1+3dx)+...+f(2)}
dx{ 1 + (1+dx)^2 + (1+2dx)^2 + (1+3dx)^2 + ...+ (1+ndx)^2}
where ndx=1 ==>n=1/dx
Now,
dx{1+1+2dx+dx^2 + 1+4dx +4dx^2 + 1+6dx + 9dx^2+...+1+2ndx+n^2 dx^2}
dx { {(n+1)*1} + 2(1+2+3+...+n)dx + (1^2 + 2^2 +3^2 +... +n^2)dx^2}

Answer 3

Looks like it is easier when f(x)=x^2

Answer 4

What about when f(x)=1/x

Answer 5

i'd have preferred to substitute 1/n in place of dx everywhere where n->inf