## leading thought

Riemann sums provide a way to solve the reputed "unsolvable problem"-- it proposes to split the area we wish to find into little, little pieces. how exactly we split it is not really a concern, because the sum of the split parts are supposedly all equal to the area in question.

there are many ways to split the area. we can cut it horizontally, we can cut it vertically, and i have even thought about drawing infinitely many circles in the area. actually, here is something that I defend my view by: think of the area as a beach. and we count the number of pebbles on the beach. perhaps it's just my unusual love for water sports...

so. usually, there are three types of Riemann sums. the left Riemann sum, the middle Riemann sum, and the left Riemann sum.

when you think of the Riemann sum, think of histograms (or bars). Because this is what Riemann sums look like, if we depict them in a graph.

the key idea is to replace the area under the function (the one that has a smooth edge) with an edgy histogram, or bar graph, that approximates the area.

there are many ways to split the area. we can cut it horizontally, we can cut it vertically, and i have even thought about drawing infinitely many circles in the area. actually, here is something that I defend my view by: think of the area as a beach. and we count the number of pebbles on the beach. perhaps it's just my unusual love for water sports...

so. usually, there are three types of Riemann sums. the left Riemann sum, the middle Riemann sum, and the left Riemann sum.

when you think of the Riemann sum, think of histograms (or bars). Because this is what Riemann sums look like, if we depict them in a graph.

the key idea is to replace the area under the function (the one that has a smooth edge) with an edgy histogram, or bar graph, that approximates the area.

the graph above gives you an idea what it's all about. the upper left one, the one with intersections on the right side of the bar, is called the right Riemann sum. and the one that had intersections in the middle of the bar, is called the middle Riemann sum. I won't even bother to tell you what is a left Riemann sum.