The area under a curve is estimated by slicing it into rectangles whose areas are easy to add.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Where the height comes from
A left sum takes each rectangle’s height from the left edge, a right sum from the right, and a midpoint sum from the centre.
Which way each errs
On an increasing function a left sum underestimates and a right sum overestimates. On a decreasing function the two swap.
Trapezoids
Joining the top corners with a slanted line gives trapezoids, which follow the curve far more closely than rectangles.
The limit
As the rectangles get thinner their total approaches the exact area. That limit is the definite integral.
What the area means
If the curve is a rate, the area is the accumulated total. Area under a velocity graph is distance travelled.
Rectangles closing in
Divide the interval into n subintervals, build a rectangle on each, and sum. As n grows and the widths shrink, the sums converge to the exact area. That limit is the definite integral, by definition.
Left, right, or midpoint
The rectangle height can be taken at the left endpoint, right endpoint or midpoint of each subinterval. For a well-behaved function all three converge to the same limit, though at different speeds.
They bracket the answer
For an increasing function, left sums underestimate and right sums overestimate. That brackets the true value between two computable numbers, which is genuinely useful when no antiderivative exists.
This is how computers integrate
Numerical integration — the trapezium rule, Simpson's rule, and their refinements — are all Riemann sums with better-chosen shapes. Most integrals in engineering are evaluated this way, not symbolically.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Second point(3, 9)
- Slope between them3
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Diego Build a Left Sum
Diego estimates the area under y = x² from 0 to 2 using two rectangles.
- Step 1
Two rectangles across a width of 2 means each is 1 wide.
Step 4: Your Turn
Practice makes it stick.
The Width
Problem 1 of 2
The interval from 0 to 6 is split into 3 rectangles. How wide is each?
The Direction
Problem 2 of 2
On an increasing function, does a left sum overestimate or underestimate? 1 over, 2 under.
Slice and Add
1 of 8
From 0 to 8 with 4 rectangles. Width of each?
2 of 8
From 1 to 5 with 8 rectangles. Width of each, as a decimal?
3 of 8
y = x² from 0 to 2, left sum with 2 rectangles. What is the estimate?
4 of 8
y = x² from 0 to 2, right sum with 2 rectangles. What is the estimate?
5 of 8
On a decreasing function, does a right sum over or under estimate? 1 over, 2 under.
6 of 8
A velocity graph. What does the area beneath it represent? 1 distance, 2 acceleration.
7 of 8
Sort each method by how it takes a rectangle height.
Tap something to move it.
- Empty
- Empty
8 of 8
From 2 to 10 with 4 rectangles. Width of each?
Step 5: Quick Check
Show what you know.
Question 1 of 2
From 0 to 12 with 6 rectangles. Width of each?
Question 2 of 2
What happens as the rectangles get thinner?
What You Learned
- A Riemann sum estimates area with rectangles.
- On an increasing function, left sums underestimate and right sums overestimate.
- The limit of these sums, as the strips thin, is the definite integral.