Question

Find the exact area bounded by f(x)=x^2 and the x-axis between x=1 and x=2 using n partitions.

Answer 1

n partitions imply a finite number...
if you're going to find the EXACT area, you'll need an infinite number of very tiny partitions...
ie, let \(\large n\rightarrow \infty \)

Answer 2

I ended up with this:
\[A = (\frac{ 1 }{ n } + \frac{ 2(1) }{ n ^{2} } + \frac{ 1^{2} }{ n ^{3} }) + (\frac{ 1 }{ n } + \frac{ 2(2) }{ n ^{2} } + \frac{ 2^{2} }{ n ^{3} }) + (\frac{ 1 }{ n } + \frac{ 2(3) }{ n ^{2} } + \frac{ 3^{2} }{ n ^{3} }) + (\frac{ 1 }{ n } + \frac{ 2(4) }{ n ^{2} } + \frac{ 4^{2} }{ n ^{3} }) +...+ (\frac{ 1 }{ n } + \frac{ 2(n) }{ n ^{2} } + \frac{ n^{2} }{ n ^{3} })\]
Simplified:
\[A = \frac{ n }{ n } + \frac{ 2 }{ n ^{2} }(1+2+3+4+...+n)+\frac{ 1 }{ n ^{3} }(1^{2}+2^{2}+3^{2}+4^{2}+...+n ^{2})\]
Is that correct? I know I have to take the limit of it eventually, but I know it simplifies more. That's what I'm unsure about.