Accumulation & Area Under a Curve — Print
Add up enough thin strips and the total is exactly the area — and usually a distance, a cost or a volume.
What the practice questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Area — Space inside, measured in square units.
- Base — The number being multiplied. In 53 the base is 5.
- Function — A rule pairing each input with exactly one output.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Left sum — Rectangles whose heights come from the left edge of each strip.
- Right sum — Rectangles whose heights come from the right edge.
- Height — The perpendicular distance from the base to the opposite corner.
A car holds a steady 30 mph for 2 hours. How far did it go?
Sixty miles, obviously. And on a speed-time graph that is a rectangle: 30 tall, 2 wide, area 60.
Now let the speed change. There is no single rectangle any more — but the graph is still there.
Chop the trip into short stretches. Over each one the speed barely changes, so each strip is almost a rectangle.
Add the strips and you have the distance, near enough. Make them thinner and "near enough" becomes exact. Area under a curve is accumulation, and the units tell you what is accumulating.
Rate times time is a rectangle
When a rate is constant, the total is rate × time — and on the graph that is an area.
30 mph for 2 hours is a rectangle 30 tall and 2 wide, and 60 is both its area and the distance.
That coincidence is not a coincidence. It is the whole idea, seen in its simplest case.
Read the units
Height in mph times width in hours gives miles. Multiplying the axis units tells you what the area means — and it is rarely area.
Strips when the rate changes
Cut the interval into n equal strips of width b-an and treat the rate as constant on each.
Take the height from the left edge or the right edge — either gives an estimate. If the function is monotonic, the two sums bracket the integral; for a function that rises and falls, that bound need not hold.
For a rising function the left sum is too small and the right sum too large.
Two estimates are better than one
For a monotonic function, if the left and right sums are 18 and 24, the true integral is between them. The gap tells you how far off you might be.
Why thinner is better
The error lives in the little triangles between each rectangle and the curve.
For a straight-line graph, halving the width quarters each little triangular error, but doubles the number of strips. The total error is halved, not quartered.
Let the width go to zero and the error vanishes, leaving the exact area.
The same move as last unit
Derivatives shrank a gap to zero to get a slope; integrals shrink a width to zero to get a total. One limit, two uses.
Notation, and what it means
∫ab f(x)dx is the exact accumulation of f from a to b.
The ∫ is a stretched S for "sum", f(x) is the height of a strip and dx is its vanishing width.
Read as a sentence: add up height times width, all the way across.
Area below the axis counts negative
A velocity graph dipping below zero means motion in the negative direction. Speed is nonnegative; integrating speed, or the absolute value of velocity, gives distance. The integral subtracts that stretch, giving displacement rather than distance travelled.
Exact areas you can already find
If the graph is made of straight lines, no calculus is needed — split it into triangles and rectangles.
A speed rising steadily from 0 to 40 over 5 seconds traces a triangle: 12(5)(40) = 100 metres.
Geometry handles every straight-edged case exactly, and it is worth checking a Riemann sum against one.
A free accuracy check
Run your strip method on a straight-line graph where you already know the answer. If it does not converge to the right number, the method is being applied wrongly.
Worked examples
A left sum
Estimate the area under f(x)=x2 from 0 to 2 using 4 left rectangles.
- Width is 2-04 = 0.5; left edges are 0, 0.5, 1, 1.5. Four strips, heights from the left.
- Heights: 0, 0.25, 1, 2.25. Sum = 3.5. Square each left edge.
- Area ≈ 0.5 × 3.5 = 1.75. The true value is 83 ≈ 2.67, so a left sum underestimates here.
1.75
Distance from a speed graph
A car speeds up steadily from 0 to 40 m/s over 5 seconds. How far does it travel?
- The speed-time graph is a triangle with base 5 and height 40. Steady change means a straight line.
- Area = 12 (5)(40). No calculus needed for a straight edge.
- = 100 metres. Same as average speed 20 times 5 seconds.
100 m
Bracketing the answer
For an increasing function on [1,3], a left sum gives 12 and a right sum 18. What can you say?
- For an increasing function the left sum is too small. Each rectangle sits under the curve.
- The right sum is too large, for the mirror reason. Each rectangle pokes above.
- So the exact area is between 12 and 18. A midpoint estimate of 15 would be a reasonable guess.
Between 12 and 18
Words to know
- Riemann sum
- An estimate of area made from rectangles under a curve.
- Left sum
- Rectangles whose heights come from the left edge of each strip.
- Right sum
- Rectangles whose heights come from the right edge.
- Definite integral
- The exact area, written ∫ab f(x)dx.
- Accumulation
- The total built up by a rate acting over an interval.
Mistakes to watch for
Using the wrong edge heights. A left sum for four strips on [0,2] takes heights at 0, 0.5, 1, 1.5 — not at 2. Including the right endpoint turns it into a different estimate.
Forgetting to multiply by the strip width. Adding the heights gives 3.5, but the area is 0.5 × 3.5 = 1.75. The sum of heights is not an area until it is given a width.
Reading the area as an area. Under a speed-time graph the units are mph times hours, which is miles. Multiply the axis units to find out what has actually been accumulated.
Ignoring negative regions. A curve below the axis contributes a negative amount, so an integral can be zero for a graph with plenty of visible area.
Assuming the left sum is always too small. That holds for an increasing function; for a decreasing one it is the other way round. Check which way the curve runs before deciding.
Archimedes found the area of a parabolic segment this way around 240 BC, two thousand years before calculus had a name.
The integral sign ∫ is a stretched letter S, for "summa". Leibniz chose it in 1675 and it has not changed since.
The fundamental theorem of calculus says that accumulating and rating are inverse operations — the two halves of this course are the same subject seen from either end.
Problem 1
A car holds a steady 30 mph for 2 hours. How many miles does it travel?
Worked answer
60
30 × 2 = 60
On the graph, that is the area of a rectangle
Problem 2
A tap runs at 4 litres per minute for 7 minutes. How many litres?
Worked answer
28
4 × 7 = 28
Problem 3
Speed rises steadily from 0 to 40 m/s over 5 seconds. Find the distance in metres.
Worked answer
100
12(5)(40) = 100
Problem 4
Speed rises steadily from 0 to 12 m/s over 6 seconds. Find the distance in metres.
Worked answer
36
12(6)(12) = 36
Problem 5
On a speed-time graph in mph against hours, what does the area represent?
- Distance in miles
- Speed in mph
- Time in hours
- Acceleration
Worked answer
Distance in miles
mph times hours gives miles
Problem 6
An interval from 0 to 2 is split into 4 equal strips. How wide is each strip?
Worked answer
0.5
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Problem 7
Estimate the area under f(x)=x2 on [0,2] with 4 left rectangles.
Worked answer
1.75
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Problem 8
Estimate the same area with 4 right rectangles.
Worked answer
3.75
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Problem 9
For an increasing function, how does a left sum compare with the true area?
- Too small
- Too large
- Exactly right
- It depends on the width
Worked answer
Too small
Every rectangle stays under the curve
Problem 10
Speed goes steadily from 10 to 30 m/s over 4 seconds. Find the distance in metres.
Worked answer
80
Average speed 20, times 4 seconds
= 80
Problem 11
A velocity graph dips below the axis. What does the integral do with that stretch?
- Subtracts it
- Adds it
- Ignores it
- Doubles it
Worked answer
Subtracts it
So it reduces the displacement
Problem 12
Estimate the area under f(x)=x on [0,4] with 4 left rectangles.
Worked answer
6
0+1+2+3 = 6
True area is 8, so this underestimates
Problem 13
Find the exact area under f(x)=x on [0,4].
Worked answer
8
12(4)(4) = 8
Problem 14
A left sum gives 12 and a right sum 18 for an increasing function. Give the midpoint of that bracket.
Worked answer
15
12+182 = 15
Problem 15
How many strips of width 0.25 fit in the interval [1, 3]?
Worked answer
8
20.25 = 8
Problem 16
A tank fills at a rate rising steadily from 2 to 8 L/min over 10 minutes. How many litres go in?
Worked answer
50
Average 5 L/min
5 × 10 = 50