Accumulation & Area Under a Curve — Challenge
Add up enough thin strips and the total is exactly the area — and usually a distance, a cost or a volume.
What these problems 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.
Problem 1
What is the area of a rectangle 8 by 5?
- 13
- 26
- 40
- 45
- 80
Worked answer
40
Area counts the unit squares that fit inside, which is the length times the width.
8· 5 = 40 square units. The perimeter would be 26.
Problem 2
A square has a perimeter of 20. What is its area?
- 20
- 25
- 40
- 100
- 400
Worked answer
25
A square has four equal sides, so each side is 204 = 5.
The area is 5· 5 = 25 square units.
Squaring the perimeter would give 400, which confuses the two measurements.
Problem 3
A rectangle has an area of 48 and a width of 6. What is its length?
- 6
- 8
- 42
- 288
- 4
Worked answer
8
Area is length times width, so 48 = 6· l.
Divide both sides by 6: l = 8.
Subtracting instead would give 42, which treats area as though it were a perimeter.
Problem 4
A shape is made of a 6 by 4 rectangle joined to a 2 by 3 rectangle, with no overlap. What is its total area?
- 24
- 27
- 30
- 36
- 42
Worked answer
30
Work out each piece separately: 6· 4 = 24 and 2· 3 = 6 square units.
Because the pieces do not overlap, the total area is the sum: 24+6 = 30 square units.
Problem 5
A rectangle's length is twice its width, and its area is 72. What is its width?
- 4
- 6
- 12
- 18
- 36
Worked answer
6
Let the width be w, so the length is 2w and the area is w· 2w = 2w2.
Setting 2w2 = 72 gives w2 = 36, so w = 6.
The length is then 12, and 6· 12 = 72 as required.
Problem 6
A car holds 12 m/s for 5 seconds, then slows steadily to a stop over the next 4 seconds. How far does it travel?
- 84 m
- 108 m
- 60 m
- 72 m
- 96 m
Worked answer
84 m
The first stretch is a rectangle: 12 × 5 = 60 m. The second is a triangle: 12(4)(12) = 24 m. Total = 84 m. Reading the area of a speed-time graph gives a distance, because m/s times s is metres.
Problem 7
Estimate the area under y = x from 0 to 6 using 3 right rectangles.
- 24
- 18
- 12
- 36
- 6
Worked answer
24
Each strip is 2 wide, with right edges at x = 2, 4, 6. The heights there are 2, 4 and 6, summing to 12. Area ≈ 2 × 12 = 24. The true area is the triangle 12(6)(6) = 18, so a right sum overestimates a rising function.
Problem 8
For y = x2 on [0,3], a left sum with three strips of width 1 gives 5 and a right sum gives 14. What can you say about the true area?
- It lies between 5 and 14
- It is below 5
- It is above 14
- It is exactly 9.5
- Nothing can be said
Worked answer
It lies between 5 and 14
The function is increasing, so each left rectangle sits under the curve and each right rectangle pokes above it. That makes 5 an underestimate and 14 an overestimate. So the true area is between them. It is in fact 9, nearer the middle than either end.