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.

Main content

# Trapezoidal sums

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)

## Video transcript

for fun let's try to approximate the area under the curve y is equal to the square root the principal root of x minus 1 between X is equal to 1 and X is equal to 6 so I want to find this entire area or I want to at least approximate this entire area and the way I'll do it the way I'll do it is by setting up five trapezoids of equal width so this will be the starting the the left boundary of the first trapezoid this will be its right boundary which will also be the left boundary of the second trapezoid this will be the right boundary of the second trapezoid this is the right boundary of the third trapezoid this will be the right boundary of the fourth trapezoid and then finally this will be the boundary of the fifth trapezoid and since we're traveling we're going from 1 to 6 so we're traveling 6 minus 1 in the X direction and I want to split it into five sections the width of each trapezoid is just going to be equal to 1 and so if we say that the width of a trapezoid is Delta X we just can now say that Delta X is equal to 1 so let's set up our trapezoids so the first trapezoid is going to look like it's going to look like that it's going to go like that actually it's going to be a triangle not really a trapezoid then the second trapezoid is going to look like this you could say a trapezoid where one of the sides as length 0 turns into a triangle then this the third trapezoid is going to look like this and then the fourth trapezoid is going to look like that and then finally you have the fifth trapezoid so let's calculate the area of each of these and then we will have our approximation for the area under the curve so let's do trapezoid or I really should say triangle this first shape whatever you want to call it what is the area of that going to be well the area of a trapezoid and you'll see this will just this will just turn into the area of a triangle it's the average of the heights of the two sides of the trapezoid the way we've looked at or you could say the average of the heights of the two parallel sides I guess way to say it so f of one that's the height here plus F of 2 plus F of 2 all of that over 2 and then we're going to multiply it times our Delta X we're going to multiply it times our Delta X actually let me do that in that same red color to show you that this is the area of that first trapezoid of that first trapezoid so times Delta X and as you see right over here if you if you look at it the F of 1 is just going to be 0 so you're gonna have F of 2 times so it's going to be that this height times this space times 1/2 which does which is just the area of a triangle but let's look at the second trapezoid trapezoid 2 right over here what is its area going to be well it's going to be F of 2 it's going to be F of 2 plus F of 3 F of 2 is this height F of 3 is this height so we're taking the average of those two Heights divided by 2 that's the average of those two Heights times the base times Delta X and then let's do trapezoid 3 I think you're getting the idea here trapezoid 3 it's going to be F of 3 plus F of 4 divided by 2 times Delta X and then let's see I'm running out of colors this is trapezoid 4 right over here so plus F of 4 plus F of 5 all of that over 2 times Delta X and then we have our last trapezoid which I will do in yellow so this is trapezoid number 5 scroll down a little bit get some more real estate so it's going to be plus I'll just write the plus over here plus F of 5 plus F of 6 over 2 times our Delta X so let's see how we can simplify this a little bit all of these terms we have a 1/2 Delta X so let's actually factor that out so remember this is our approximation of our area so area area is approximately remember this is just a rough approximation it's very clear actually you might say this is pretty good using the trapezoids but there are there it is pretty clear or that we are letting go of some of the area we're letting go of that area we're letting go of some of this right over here you can barely see it some of this right over here you can barely see it but we are this is going to be it looks like an underestimate but it is a decent approximation let's see if we can simplify this expression so let's go approximately going to be equal to I'm going to factor out a delta x over two I'm going to factor out a delta x over two and then what I'm left with and I will switch to a neutral color if i factor out a delta x over two then I have just an F of one F of one and then I have two F of twos so plus 2 times F of 2 and I'm doing this because you might see a formula that looks something like this in your calculus book and it's not some mysterious thing they just really summed up the areas of the trapezoids and then we're going to have to F of threes plus 2 times F of threes plus we're going to have to F of fours plus 2 times F of 4 and then we're going to have to F of 5s plus 2 times F of 5 and then finally we're going to have 1 F of 6 plus F of 6 if you were to generalize it you have one of the first endpoint the function evaluated the first endpoint one of it at the very last endpoint and then two of all of the rest of them but this is just the area of trapples I'm not actually a big fan when textbooks write this because you don't when you see this it's hard to visualize the trapezoids when you see this it's much clearer how you might visualize that but with that out of the way let's actually evaluate this lucky for us the math is simple our Delta X is just 1 our Delta X is just 1 and then we just have to evaluate all of this business F of 1 let's just remind ourselves what our original function was our original function was the square root of x minus 1 square root of x minus 1 so f of 1 is the square root of 1 minus 1 so that is just going to be 0 this is this expression right over here is going to be 2 times the square root of 2 minus 1 the square root of 2 minus 1 is just 1 so this is just going to be 2 actually let me do it in that same well I'm now using the purple for a different purpose than just the first trapezoid hopefully you realize that just sticking with that pin color then f of 3 3 minus 1 is 2 square root of 2 so the function evaluated 3 is the square root of 2 so this is going to be 2 times the square root of 2 then the function evaluated for when you evaluate it for you get the square root of 3 so this is going to be 2 times the square root of 3 and then you get 2 times the square root of 4 5 minus 1 is 4 2 times the square root of 4 is just 4 and then finally you get F of 6 is square root of 6 minus 1 is the square root of 5 square root of 5 and I think we are now ready to evaluate so let me get my handy ti-85 out and calculate this so it's going to be well I'm just going to calculate what I'll just multiply it so 0.5 times open parenthesis well it's a 0 I'll just write it just to you know what I'm doing 0 plus 2 whoops lost my calculator plus 2 times the square root of 2 plus 2 times the square root of 3 plus 4 plus 4 I'm almost done plus plus the square root of 5 so let me write that plus the square root of 5 gives me now we are ready for our drum roll it gives me and I'll just round it 7 point 2 6 so the area is approximately equal to approximately equal to 7 point to 6 under the curve under the curve Y is equal to the square root of x minus 1 between x equals 1 and x equals 6 and we did this using trapezoids
AP® is a registered trademark of the College Board, which has not reviewed this resource.