Accumulation & Area Under a Curve — Drills
Accumulation & Area Under a Curve has 6 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Rectangle
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A steady rate of 9 litres per minute runs for 6 minutes. How many litres?
Worked answer
54
9 × 6 = 54
Problem 2
A steady rate of 18 mph runs for 4 hours. How many miles?
Worked answer
72
18 × 4 = 72
Triangle
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Speed rises steadily from 0 to 20 m/s over 6 seconds. Find the distance in metres.
Worked answer
60
12(6)(20) = 60
Problem 2
Speed rises steadily from 0 to 30 m/s over 5 seconds. Find the distance in metres.
Worked answer
75
12(5)(30) = 75
Trapezium
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Speed goes steadily from 2 to 20 m/s over 6 seconds. Find the distance in metres.
Worked answer
66
Average = 2+202 = 11
11 × 6 = 66
Problem 2
Speed goes steadily from 4 to 16 m/s over 4 seconds. Find the distance in metres.
Worked answer
40
Average = 4+162 = 10
10 × 4 = 40
Width
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
The interval from 2 to 8 is split into 10 equal strips. How wide is each strip?
Worked answer
0.6
610 = 0.6
Problem 2
The interval from 3 to 9 is split into 5 equal strips. How wide is each strip?
Worked answer
1.2
65 = 1.2
Strips
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many strips of width 0.5 fit in an interval of length 2?
Worked answer
4
20.5 = 4
Problem 2
How many strips of width 0.25 fit in an interval of length 4?
Worked answer
16
40.25 = 16
Leftsum
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Estimate the area under f(x)=x on [0,5] using 5 left rectangles of width 1.
Worked answer
10
Heights 0, 1, 2, 3, 4
Sum = 10, and each width is 1
True area is 12.5, so this underestimates
Problem 2
Estimate the area under f(x)=x on [0,4] using 4 left rectangles of width 1.
Worked answer
6
Heights 0, 1, 2, 3
Sum = 6, and each width is 1
True area is 8, so this underestimates