Lesson video

Hello, my name is Mr Clasper.

And today we are going to be estimating the area under a curve.

How can we estimate the area under the curve between where X is equal to zero and where X is equal to four? What we could do is to create a trapezium which has parallel sides and a perpendicular height of four.

I'll then need my area of a trapezium formula.

Once I've done this, before I find the pieces of information which I need, so my two parallel sides are my base, I can substitute these into the formula.

This would give me an area of 60 units squared.

Do you think this is an underestimate or an overestimate? The answer of 60 units squared would be an overestimate for the area under the curve between the two given points.

This is because part of the area of my trapezium lies above on the graph.

How can we make our answer more accurate? In the previous example, we used one strip or one trapezium.

In this example, we can use two strips or two trapeziums. So if we can find the area of each of these two trapeziums and add them together, this will give us a more accurate answer.

So we're going to add the area of the first trapezium and the area of the second trapezium to find my total area.

I can substitute the information in that I know, and this will help me with my calculations.

This would give me an answer of 52 units squared.

Can you see how this is closer to the exact area? So our two trapeziums have a smaller amount of their area which lies above the graph, compared with the first example.

Can we make this even more accurate? Let's try this by using four strips or four trapezium.

We need to find the lengths of each of our parallel sides.

And we can see that each of my four trapeziums have a perpendicular height of one unit.

So if we label these A, B, C, and D, we could find the area of A, and we could find the area of B.

We could find the area of C, and then find the area of D.

And my total area will be the sum of all four areas.

So the total estimate for the area would be 50 units squared.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here is your solution.

So remember if we're using one strip, it means we're drawing one trapezium, and we need to calculate the area of the trapezium.

When we do this, we should find that the area is approximately 52 units squared.

And here is part b.

Pause the video to complete your task and click resume once you're finished.

And here is your solution for part b.

So this time we were asked to use two strips.

So if we use two trapeziums, each of the trapeziums will have a perpendicular height of two.

And if we calculate each of those and find the sum, that means we have a total of 44 units squared.

Here is part c.

Pause the video to complete your task and click resume once you're finished.

And here is your solution for part c.

So once again, we're asked for four strips.

So this means we're using four trapeziums and we need to find the area of each of the trapeziums and then find the sum to get our total area.

They should give us 42 units squared.

Here as part d.

Pause the video to complete your task and click resume once you're finished.

And here is your solution to part d.

So our answers were over estimates.

And this is because in each example, we had parts of the area of some of our trapeziums, which were above the graph.

Therefore it's an overestimate.

And here is your last question.

Pause the video to complete your task, click resume once you're finished.

And here is your solution.

So using the method that we have used before, if you use more than one strip, your answer will be more accurate.

You should eventually get an area of 265.

And as we know, on a speed-time graph, the area underneath the graph represents our distance.

Therefore, the final answer was approximately 265 metres.

And that brings us to the end of our lesson.

So you've been estimating the area under a curve.

Why not show off your skills with our exit quiz? I'll hopefully see you soon.