Main content

### Course: AP®︎/College Calculus BC > Unit 6

Lesson 2: Approximating areas with Riemann sums- Riemann approximation introduction
- Over- and under-estimation of Riemann sums
- Left & right Riemann sums
- Worked example: finding a Riemann sum using a table
- Left & right Riemann sums
- Worked example: over- and under-estimation of Riemann sums
- Over- and under-estimation of Riemann sums
- Midpoint sums
- Trapezoidal sums
- Understanding the trapezoidal rule
- Midpoint & trapezoidal sums
- Riemann sums review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Riemann approximation introduction

Approximating the area under a curve using some rectangles. This is called a "Riemann sum". Created by Sal Khan.

## Want to join the conversation?

- Hey Sal, I have a question. Are Riemann sums actually useful for anything once you know how to integrate? It seems a lot easier to just learn integration.(90 votes)
- Riemann sums are useful when we are dealing with real world data, but we don't know the exact pattern it is following. To integrate, you need to know the function, but you can use Riemann sums as an approximation whenever you know some of the data points.(69 votes)

- With my limited knowledge of calculus, the Riemann sums seem pretty inefficient, since you'll always have some space not covered by the rectangles. Is there a way that you can get an exact answer, or is this the best that has been come up with at this time? Of course, I don't know much in calculus yet, since I've only done Alg. 1&2, Geo., and PreCal, so there might be a construct that I don't know of that would answer my question. Thanks to any who will answer this! :)(10 votes)
- You need to do differential calculus before integral calculus.

But, to answer your question, yes there is a way to get an exact answer. It is what happens when you split the area up into infinitely many rectangles of infinitesimally small widths. That way, you have no error at all. It is called a definite integral.

But, you really do need to master differential calculus first or you will be lost.(30 votes)

- Why did Sal construct 4 rectangles? Why not 5, 6 etc... or is 4 rectangles given in the problem?(9 votes)
- Most of the practice problems of integral calculus asked for 4 equal subsections so I believe it was a choice made by him in this example. Just like you mentioned it is oftentimes already given in the problem. With more rectangles you get a better approximation of the function.(12 votes)

- How do you know how many pieces you're suppose to split it in? Like what number are you suppose to divide b-a by?(6 votes)
- The number of pieces you use is a function the accuracy (how close to the actual value of the area) you want.

While this technique has it's place in numerical analysis, you are going to see that as the number of pieces you split the interval into approaches infinity, the value of the result of the sum approaches the actual value of the area - and that this is the foundation for the concept of integration - which is the next section of this track.(10 votes)

- Wouldn't finding the over-estimate and the under-estimate of the curve, adding them together, and dividing by two give a more accurate estimate of the area? Though it still wouldn't be completely accurate.(6 votes)
- Yes, that's equivalent to approximating the area with trapezoids, which is covered later.(8 votes)

- At0:20seconds, how do you know that you need four rectangles?(3 votes)
- We can choose how many rectangles we like.

The approximation will improve with more rectangles.

The best approximation comes with infinite rectangles.

This can be done by using infinitesimal width on the rectangles.

I hope this was a little helpful!(9 votes)

- If the area under the curve is below the x-axis, would it have a negative value?(5 votes)
- Is there a way to find the area above the function up to a certain range?(3 votes)
- Yes, you'd have to define the upper bound with a function of some sort, then apply the same integral peocess.(3 votes)

- Are Riemann sums the ones using left endpoints because I asked this question in a different video and im not sure if this video is the one that has now answered my question.(3 votes)
- Riemann sums can have a left, right, middle, or trapezoidal approximations. The most accurate are usually the trapezoidal and middle rectangle approximations because they only give up a small amount of area. However, Riemann sums will usually give more accurate approximations based on the number of rectangles and trapezoids; for example, an approximation using 4500 left rectangles will be better than simply using four rectangles to express the area under a curve. Similarly, an infinite number of rectangles will have a more accurate approximation of the area instead of simply using 4500 rectangles.(2 votes)

- In physics I have seen that often in deriving equations the sigma sign(for addition) is replaced in the subsequent steps by the integral sign. Can someone please tell me more about this(2 votes)
- The ∑ becomes ∫ when there are infinitely many, infinitesimally thick rectangles. Thus, the ∑ indicates you are adding up a finite number of rectangles while the ∫ indicates you are adding up infinitely many rectangles.(3 votes)

## Video transcript

What we're going to
try to do in this video is approximate the
area under the curve y is equal to x squared plus 1
between the intervals x equals 1 and x equals 3. And we're going
to approximate it by constructing four rectangles
under the curve of equal width. So let's first think about what
those rectangles look like, so four rectangles of equal width. So it looks like that,
like that, and like that. And I haven't really defined the
top of the rectangles just yet. But let's think about
what those widths have to be if they're going
to be equal width. And we can call
that width delta x. So this distance
right over here, we're going to
call that delta x. So delta x is going to have
to be the total distance that we're traveling in x. So we finish at 3. We started at 1. And we want four
equal-width rectangles. So it's going to
be equal to 1/2. So for example,
this first interval between the boundary between the
first rectangle and the second is going to be 1.5. Then we go 1/2 to 2. Then we go to 2.5. And then we go 1/2 to 3. Now, let's think
about how we'll define the height of the rectangles. For the sake of this video--
we'll see in future videos that there's other
ways of doing this-- I'm going to use the left
boundary of the rectangle to define the height-- or
the function, I should say. I'm going to use the function
evaluated at the left boundary to define the height. So for example, for
the first rectangle, this point right
over here is f of 1. And so I will say that
that is the height of our first rectangle. Then we go over here
to the left boundary of the second rectangle. We're now looking at the
function evaluated at 1.5. So that is f of 1.5. That's the height. And so we get our
second rectangle. Then, we get-- I can keep
going like this-- we get, for this third
rectangle, we have the function evaluated at 2. So that's right over here. That's f of 2. And so then we get
our third rectangle. And then, finally, we
have our fourth rectangle, the function evaluated at 2.5. So the function evaluated
at 2.5 is the height. So this is f of 2.5. Remember, in each
of these, I'm just looking at the left
boundary of the rectangle and evaluating
the function there to get the height
of the rectangle. Now that I've set
it up in this way, what is the total
approximate area using the sum of
these rectangles? And clearly, this isn't going
to be a perfect approximation. I'm giving up on a
bunch of area here. Let me see if I can color
that in with a color that I have not used. So I'm giving up. I'm giving up this area. I'm giving up this area. I'm giving up that area. I'm giving up that area there. But this is just
an approximation, and maybe if I had
many more rectangles, it would be a better
approximation. So let's figure out what the
areas of each of the rectangles are. So the area of this
first rectangle is going to be the
height, which is f of 1, times the base,
which is delta x. The area of the
second rectangle is going to be the height, which
we already said was f of 1.5, times the base, times delta x. The height of the
third rectangle is going to be the function
evaluated at its left boundary, so f of 2-- so plus f of 2
times the base, times delta x. And then, finally, the area of
the third rectangle, the height is the function
evaluated at 2.5, so plus-- that's
a different color than what I wanted to use. I wanted to use that
orange color-- so plus the function evaluated
at 2.5 times the base. This is going to be equal
to our approximate area-- let me make it clear--
approximate area under the curve, just the
sum of these rectangles. So let's evaluate this. So this is going
to be equal to f of-- it's going to be equal to
the function evaluated at 1. 1 squared plus 1 is just 2, so
it's going to be 2 times 1/2. Plus the function
evaluated at 1.25. 1.25 squared is 2.25. And then you add 1 to
it, it becomes 3.25. So plus 3.25 times 1/2. And then we have the
function evaluated at 2. Well, 2 squared plus 1 is
5, so it's 5 times 1/2. And then finally, you have
the function evaluated at 2.5. 2.5 squared is 6.25 plus 1. So that's 7.25 times 1/2. And just to make
the math simpler, we can factor out the 1/2. So this is going to be
equal to-- write 1/2 in a neutral color-- 1/2
times 2 plus 3.25 plus 5 plus 7.25, which is equal
to 1/2 times-- let's see if I can do this in my head. 2 plus 5 is easy. That's 7. 3 plus 7 is 10. And then we have 0.25
plus 0.25, so it's going to be 10.5 plus 7 is 17.5. So 1/2 times 17.5,
which is equal to 8.75, Which, once again, gives
us an approximation. And clearly, the way I've
drawn it right over here, for the function
we're using, it's going to be an underestimate,
because we've given up all that pink area that
I had colored in before. It's an underestimate,
but it's an approximation of the area under the curve. In the next few
videos, we're going to try to generalize
this to situations where we have an arbitrary
function and we have an arbitrary
number of rectangles. And we'll also start--
in videos after that, we'll look at rectangles
where we define the height not by the left boundary, but
by the right boundary, or by the midpoint. Or maybe we don't use
rectangles at all. Maybe we might use
things like trapezoids. Anyway, have fun.