riemann sums ap calculus ab multiple choice conceptual questions

Understand the information the question gives you For the purposes of simplicity, I am going to demonstrate a Left Riemann Sum.ġ. So imagine you are given this equation: f(x) = x^2. If we start at k = 0, we have a left Riemann sum. In this instance, we're looking at the right x value of each. 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.

Riemann sums ap calculus ab multiple choice conceptual questions