I have this questions about Reimann Integration using a given geometric partition with an unknown value of n for the function $f(x) = 1/x^3$ . enter image description hereI have attached the work I have done so far but I don't understand where I have gone wrong. Thanks.
f(x) = 1/x^3 over [2,3] so a=2 and b=3 $$Q_n$$ = $${2,2\eta,2\eta^2,2\eta^3.....2\eta^n}$$ So $\eta^n$=3/2 So a subinterval= [2$\eta^(i-1)$,2$\eta^i$] $$M_i$$ = lub f(x)= $$1/2\eta^(i-1)$$ and glb = $$m_i$$ = $$(1/2\eta^i)^3$$ Simplifying $$m_i$$ gives me $$(1/12\eta)$$ and simplifying $$M_i$$ gives me $$((\eta)^3)m_i$$ Using this i get the integral = 1/12 which is wrong, can anyone help with where it is wrong?
The function you're interested in is $ f(x)=\frac{1}{x^3} $, so since this is decreasing, you should have that $ M_{i}=\frac{1}{(2{\eta^{i-1}})^3} $ and that $ m_{i}=\frac{1}{(2{\eta^{i}})^3} $.
You can start by finding $ L(Q_n,f) = \sum_{i=1}^n m_i(x_i-x_{i-1}) $.
Hint: Take all possible factors out of the sum, then use formula for the sum of the first n terms of a geometric series.
Then notice that $ M_i = \eta^3m_i $, so you can use this and the expression for $ L(Q_n,f) $ to easily find $ U(Q_n,f) $.
I assume that you'll be fine from here, but after considering both $ \lim_{n \to \infty} L(Q_n,f) $ and $ \lim_{n \to \infty} U(Q_n,f) $, you see they both tend to the same desired result.