Upper & Lower Sums: Calculating & Understanding Mi & mi

[imull]

I am really having trouble understanding how upper and lower sums are calculated. In the equations Ʃf(M_i)Δx and Ʃf(m_i)Δx, what do M_i and m_i represent?

 

[jbunniii]

It's easiest to understand if you look at a picture. In the figure below (shamelessly borrowed from elsewhere via Google Images), the pink rectangles represent the upper sums and the green rectangles represent the lower sums.

In other words, the $M_i$ are the heights of the taller (pink + green) rectangles, and the $m_i$ are the heights of the shorter (green only) rectangles.

How are the heights determined? Quite simple. Each tall rectangle is exactly as high as the maximum value of the function within the width of the rectangle. Each short rectangle is exactly as high as the minimum value of the function.

 

[pwsnafu]

In other words, the $M_i$ are the heights of the taller (pink + green) rectangles, and the $m_i$ are the heights of the shorter (green only) rectangles.

No. $M_i$ are the x values such that $f(M_i)$ are the heights of the taller rectangles. Similarly for $m_i$.

 

[LCKurtz]

You're correct of course, but that is such unusual notation it makes you wonder if the OP copied the expression correctly from his text.

 

[jbunniii]

No. $M_i$ are the x values such that $f(M_i)$ are the heights of the taller rectangles. Similarly for $m_i$.

Oh, weird. You're right. I misread the question to say what it would have said in most books.

 

[imull]

I did copy them from my text. I wasn't sure how else to go about this.