# Partition of Interval

• Jan 22nd 2010, 06:47 AM
WartonMorton
Partition of Interval
Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then...

Uf(P) - Lf(P) = [f(a) - f(b)]▲x
• Jan 22nd 2010, 08:05 AM
Drexel28
Quote:

Originally Posted by WartonMorton
Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then...

Uf(P) - Lf(P) = [f(a) - f(b)]▲x

Uhh? What are these?I'm not sure I agree with the question. But, I don
t know what these are? Is $\displaystyle U\left(P,f\right)=\sum_{j=1}^{n}\sup_{x\in[x_j,x_{j+1}]}f(x)\text{ }\Delta x_j$?
• Jan 22nd 2010, 09:52 AM
Plato
Quote:

Originally Posted by WartonMorton
Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then... Uf(P) - Lf(P) = [f(a) - f(b)]▲x

Notice that this is a regular partition. So $\displaystyle \Delta_x=\frac{b-a}{n}$.
Also this is a decreasing function which means $\displaystyle Uf(P) = \sum\limits_{k = 0}^{n - 1} {f(x_k )\Delta_x }~\&~ Lf(P) = \sum\limits_{k = 1}^n {f(x_k )\Delta_x }$.
Can you see how to finish?