xk, therefore, is a way of calculating the x value of each rectangle. k is what rectangle we're on and Δx is the width of each rectangle. a is our initial value, our starting point on our interval. You'll see in the formula above that xk = a + kΔx. Well, how do we find f(xk)? First, we must find xk itself. So for example, at k = 1, we are looking at the first rectangle. What is k, though? k tells us essentially what rectangle we're on.
If Δx is the width of our rectangles, f(xk) must be the height! This makes pretty intuitive sense since, for some x value, f(x) tells you essentially the height of the graph at that x. This makes sense, we took a range of 8 and cut it into 4 equal rectangles (The rectangles in a Riemann sum will have the same width if you are just given a number of rectangles and an interval). We have n = 4 rectangles, so Δx = (b - a)/n = 8/4 = 2. In this example, we are finding the sum from 0 to 8, so b - a = 8.
Therefore, (b-a)/n is the equivalent of saying we're taking our interval and chopping it up into n sized pieces on that interval. Similarly, n is the number of rectangles we have (in a perfect Riemann sum, n approaches infinity). For example, if we were finding the area on the interval, b - a = 5. What does this mean? Our area is being taken on the interval, so b-a is just the length of our interval. You can see in the image that Δx = (b-a)/n.
So one of these must be the height of our rectangles, the other must be our width. Recall that A = bh (or alternatively A = lw) for rectangles. Remember, we're summing together the areas of rectangles, so this must be the area of some rectangle.
Now, let's tackle the inside of the sum, the f(xk)*Δx. The use of sigma here makes sense since we're summing together a bunch of rectangles. The large E looking symbol, if you are unfamiliar, is the Greek letter sigma and is used for summing together a bunch of elements or numbers. Woah! That looks really really scary, doesn't it? Well, it's actually quite simple, so let's break it down.
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