If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Calculus, all content (2017 edition)>Unit 4

Lesson 11: Definite integral as the limit of a Riemann sum

# Ranking area estimates

A good way to test your understanding of Riemann sums is to rank the values of various different sums.

## Want to join the conversation?

• At 3.33 you move on to the next expression, but you use right handed rectangles, what determined that it was left or right? The only difference in the expression was the starting point of N....It is stated that because it is multiplied by one, but so is the previous expression.

Thanks!
• Right and left handed rectangles can be determined by looking at the index in the summation. Notice that the first example starts at i=0. This means that we start the Riemann sum at x= -5, and move forward in 1 unit increments to the right. This is obviously a left handed sum. Now look at example 2. We start the index at i=1, which means the first x value evaluated in f(x) is x= -4. This means that this is a right handed sum. If you ever have issues determining which side the rectangles are based on, try the first few values in the summation.
• why is the definite integral equal to the exact area under the curve ? isnt it the sum of infinitely small rectangles ?Aren't we either way giving up infinitely small piecesof area?
• As you just say, we are either overestimating or underestimating by an "infinitely small" margin, that means that the error is infinitely small, so for all practical purposes the error is 0 and the area under the curve is exact.
• are right handed rectangles always overestimate and left handed underestimate?
• No. it depends on the shape of the curve. Look at the estimates for f(x) = e^(-x)
• How did you know the first expression wanted left-handed rectangles vs right-handed rectangles?
• Because the first one, the "i" starts with 0, which means the first height we will consider is f(-5). f(-5) is on the left side of the rectangle, I hope you can see it.
On the other hand, on the second expression, "i" starts with 1, which means our first height is f(-5+1) which equals to f(-4). f(-4), as you can see, is the right side of the rectangle.

Furthermore, when "i" equals 10 on the second expression, our height is f(5), the right side height of our last rectangle. Similarly, on the first expression, when "i" equals 9, we consider the height as f(4), the left side. Hope you understand it.
• So, the video is about ranking area estimates, and at the end he ranks them from largest to smallest, but he never ranks them by accuracy. Obviously the definite integral is the most accurate, and the i --> 20 is the next most accurate, but is there any general rule, perhaps based on concavity that allows us to judge left vs right handed rectangles in terms of accuracy?
• As you noticed, accuracy is more a function of subdivisions than right handed or left handed rectangles. Since functions can be unpredictably curvy, one rule may be more accurate than another (then assuming the same number of subdivisions). Only the definite integral is guaranteed to be exact all the time.
• Does whether it is a right handed or left handed rectangle depend on the equation? Could the same expression that you would consider left handed rectangles on one graph be right handed rectangles on another? Thanks, and will vote up an acceptable answer.
• In this video, Sal is trying to find an estimate of the blue area. If you see the first summation, you will need to consider the values f(-5), f(-4),...,f(4). If you find the right handed area at x= -5 i.e. f(-5)*delta(x) gives the area of the rectangle right of x= -5, then you would be calculating outside of the blue area. Therefore, left-estimate was better choice here.
Similarly for the second summation, if you find the left handed area at x=4 i.e. f(4)*delta(x) gives the area of the rectangle right of x=4, then you would again be calculating outside of the blue area. So right handed rectangles are used.
Hope this helps!
• Couldn't we just take the average of the right-handed and the left-handed to make the estimated area way more accurate? Just thinking.
• A lot of times this does help, but it most cases it is still inaccurate. Imagine trying to do this with a very tight sine wave, or a very zig-zaggy curve. The estimates might come out very different. Sometimes the right-handed estimate will be more positive then the left-handed estimate is negative, and then you will get slightly inaccurate results.
• what handed is the last equation actually? right handed I guess? because for it to be left-handed, It needs to start right from -5 which can be done only by saying from 0 to infinite. But when we expand the last equation is it from 0 to infinite or 1 to infinite?