Accumulation & Area Under a Curve — Practice
Add up enough thin strips and the total is exactly the area — and usually a distance, a cost or a volume.
What these 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.
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