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

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- Jan 22nd 2010, 06:47 AMWartonMortonPartition 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 AMDrexel28
- Jan 22nd 2010, 09:52 AMPlato
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?