## definition

Hopefully, by now, you have read the 'leading thought' section. now. You are prepared for the precise definition of Riemann sums.

the definition is something that looks like this:

the definition is something that looks like this:

now, what does this thing mean? going over this bulk tells us rather slowly that this is just an operation that goes from one side of the graph to the other, and gradually draw a histogram that approximates the area beneath the curve.

or, in other words, what I have been trying so hard to say in the leading thought part.

Now, remember that we have learned about limits, and as calculus is the study of limits, we expect Riemann sums can connect someway to limits. Here is how (for now, you can ignore the sign on the left):

or, in other words, what I have been trying so hard to say in the leading thought part.

Now, remember that we have learned about limits, and as calculus is the study of limits, we expect Riemann sums can connect someway to limits. Here is how (for now, you can ignore the sign on the left):

what the above expression means is just a simple operation. consider the bar graph we used to approximate the function. Of course, each bar has a width. What if the width is zero? Than the sum of the areas of those "infinitely thin" bars would be equal to the exact area under the curve.

The problem remains how to implement this criterion (zero width). The expression above provides an excellent solution.

Consider appropriating the area with infinitely many bars. the width of each bar would be something (the width of the area beneath the curve) divided by infinity, which is zero. That is because any constant, divided by infinity, is zero. So, this is equal to approximating the area with a collection of infinitely many lines beneath the curve.

And the length of a line has a mathematical significance: it is the function value at that point. So the sum of all the lengths would be equal to adding up every value-- that's what it does!

And with that, mission accomplished. All that is left to do is to evaluate the limit.

Such a calculation is often messy. That's why you need to look at the next section, fundamental theorems in calculus.

The problem remains how to implement this criterion (zero width). The expression above provides an excellent solution.

Consider appropriating the area with infinitely many bars. the width of each bar would be something (the width of the area beneath the curve) divided by infinity, which is zero. That is because any constant, divided by infinity, is zero. So, this is equal to approximating the area with a collection of infinitely many lines beneath the curve.

And the length of a line has a mathematical significance: it is the function value at that point. So the sum of all the lengths would be equal to adding up every value-- that's what it does!

And with that, mission accomplished. All that is left to do is to evaluate the limit.

Such a calculation is often messy. That's why you need to look at the next section, fundamental theorems in calculus.