Trapezoidal rule: IB Mathematics: Applications & Interpretation questions with answers
Free Trapezoidal rule questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.
A pond is 50 m long. Its width, w metres, is measured every 10 m along its length.
Distance (m)
0
10
20
30
40
50
Width (m)
0
13
9
17
13
0
Table 1.1
(a)
Use the trapezoidal rule to estimate the surface area of the pond.
[3]
(b)
The pond has a constant depth of 0.8 m. One litre of treatment is needed for every 40 m3 of water. Find the number of litres needed, and explain whether this is likely to be enough.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
520 m2
h = 10 (A1) trapezoidal rule (M1) 520 m2A1
3
(b)
11 litres
volume 416 m3(A1) ÷ 40 (M1) 11 litres (round up) A1 the edges are curved outwards, so the rule may underestimate; buy a little extra R1
4
Question 2Structured · 6 marks
The table shows values of f(x) = 10x + 1, correct to four significant figures.
x
0
1
2
3
4
f(x)
10
5
3.333
2.5
2
Table 2.1
(a)
Use the trapezoidal rule with 4 intervals to estimate ∫04f(x)dx.
[3]
(b)
Find the percentage error in this estimate.
[2]
(c)
State, with a reason, whether the estimate is an overestimate or an underestimate.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
16.8
h = 1 (A1) 12(10.00 + 2(…) + 2.000) (M1) 16.8 A1
3
(b)
4.59%
exact value 16.1 from GDC (M1) 4.59% A1
2
(c)
overestimate
overestimate, the curve is concave up so the trapezoids lie above it R1
1
Question 3Structured · 7 marksA* challenge
A pond is 25 m long. Its width, w metres, is measured every 5 m along its length.
Distance (m)
0
5
10
15
20
25
Width (m)
0
10
19
14
12
0
Table 3.1
(a)
Use the trapezoidal rule to estimate the surface area of the pond.
[3]
(b)
The pond has a constant depth of 1.2 m. One litre of treatment is needed for every 40 m3 of water. Find the number of litres needed, and explain whether this is likely to be enough.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
275 m2
h = 5 (A1) trapezoidal rule (M1) 275 m2A1
3
(b)
9 litres
volume 330 m3(A1) ÷ 40 (M1) 9 litres (round up) A1 the edges are curved outwards, so the rule may underestimate; buy a little extra R1
4
Question 4Structured · 6 marks
The table shows values of f(x) = 10x + 1, correct to four significant figures.
x
0
0.5
1
1.5
2
f(x)
10
6.667
5
4
3.333
Table 4.1
(a)
Use the trapezoidal rule with 4 intervals to estimate ∫02f(x)dx.
[3]
(b)
Find the percentage error in this estimate.
[2]
(c)
State, with a reason, whether the estimate is an overestimate or an underestimate.
overestimate, the curve is concave up so the trapezoids lie above it R1
1
Question 5Structured · 7 marksA* challenge
A pond is 25 m long. Its width, w metres, is measured every 5 m along its length.
Distance (m)
0
5
10
15
20
25
Width (m)
0
8
11
17
20
0
Table 5.1
(a)
Use the trapezoidal rule to estimate the surface area of the pond.
[3]
(b)
The pond has a constant depth of 0.8 m. One litre of treatment is needed for every 40 m3 of water. Find the number of litres needed, and explain whether this is likely to be enough.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
280 m2
h = 5 (A1) trapezoidal rule (M1) 280 m2A1
3
(b)
6 litres
volume 224 m3(A1) ÷ 40 (M1) 6 litres (round up) A1 the edges are curved outwards, so the rule may underestimate; buy a little extra R1
4
Question 6Structured · 6 marks
The table shows values of f(x) = √x + 1, correct to four significant figures.
x
0
0.5
1
1.5
2
f(x)
1
1.225
1.414
1.581
1.732
Table 6.1
(a)
Use the trapezoidal rule with 4 intervals to estimate ∫02f(x)dx.
[3]
(b)
Find the percentage error in this estimate.
[2]
(c)
State, with a reason, whether the estimate is an overestimate or an underestimate.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in