SL 5.8—Trapezoid rule Questionbank

Practice SL 5.8—Trapezoid rule with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

A structural engineer is designing a curved roof for a building. The height of the roof, $h(x)$ , in meters, is modeled by $h(x)=-0.02 x^{3}+0.3 x^{2}+0.4 x$ , for $0 \leq x \leq 10$ , where $x$ is the horizontal distance from one end. The cross-sectional area under the roof is critical for ventilation calculations. The engineer takes measurements at specific points:

$x(\mathrm{~m})$	0	2	4	6	8	10
$h(x)(\mathrm{m})$	0	0.88	1.92	2.52	2.56	2.00

Find $\frac{d h}{d x}$ and determine the maximum height of the roof.

[4]

Use the trapezoidal rule with 5 intervals to estimate the cross-sectional area under the roof.

[3]

Sketch the graph of $h(x)$ from $x=0$ to $x=10$ and shade the region corresponding to the estimate in part (b).

[2]

Write down the integral representing the exact cross-sectional area and evaluate it.

[3]

Calculate the percentage error of the trapezoidal rule estimate compared to the exact area.

[2]

Explain why the trapezoidal rule estimate differs from the exact area, referencing the shape of the curve.

[2]

Question 2

SLPaper 2

A water tank's cross-sectional area, $A(h)$ , in $\mathrm{m}^{2}$ , varies with height $h$ , in meters. The volume is $\int_{0}^{8} A(h) d h$ . Assume $A(h)=a+b h+c h^{2}$ , with measurements:

$h(\mathrm{~m})$	0	2	4	6	8
$A(h)\left(\mathrm{m}^{2}\right)$	20	18	18	20	24

Use the trapezoidal rule with 4 intervals to estimate the volume.

[3]

Sketch the graph of $A(h)$ and shade the region used in part (a).

[2]

Use the points to form a system of equations for $a, b, c$ . Solve for one coefficient.

[2]

Find the exact volume by integrating the quadratic model, assuming $A(h)=24- 0.5 h^{2}+h .

[3]

Calculate the percentage error of the trapezoidal estimate.

[2]

If water flows in at $2 \mathrm{~m}^{3} / \mathrm{min}$ , estimate the filling time using the trapezoidal volume.

[2]

Explain how the number of intervals affects the trapezoidal rule's accuracy.

[2]

Question 3

SLPaper 1

A roof's height, $y(x)$ , in meters, is modeled by $y(x)=-0.1 x^{2}+1.2 x$ , for $0 \leq x \leq 12$ , where $x$ is the horizontal distance from one end. The area under the curve represents the cross-sectional area of the roof space.

Find $\frac{d y}{d x}$ and determine the maximum height of the roof.

[3]

Use the trapezoidal rule with 4 intervals to estimate the cross-sectional area.

[3]

Compare the estimate from part (b) with the exact area found by integration.

[3]

Question 4

SLPaper 2

A sculpture's height, $s(t)$ , in meters, is modeled by a cubic function over a 9 -meter base, where $t$ is the distance from one end. Heights are:

$t(\mathrm{~m})$	0	3	6	9
$s(t)(\mathrm{m})$	1	2.5	3	2

Use the trapezoidal rule with 3 intervals to estimate the area under the curve.

[3]

Assume $s(t)=a t^{3}+b t^{2}+c t+d$ . Use the given points to find two linear equations in $a, b, c, d$ .

[2]

Sketch the graph of $s(t)$ and shade the region used in part (a).

[2]

Question 5

SLPaper 1

A scientist models the concentration of a chemical in a solution, $C(t)$ , in $\mathrm{mg} / \mathrm{L}$ , over 5 hours, using $C(t)=2 e^{0.3 t}$ . The total amount of chemical is proportional to the area under the curve. Measurements are taken hourly:

$t(\mathrm{~h})$	0	1	2	3	4
5
$C(t)(\mathrm{mg} / \mathrm{L})$ 9.07	2	2.70	3.64	4.95	6.70

Use the trapezoidal rule with 5 intervals to estimate the total amount of chemical.

[3]

Show that $\ln C(t)=0.3 t+\ln 2$ .

[2]

Using part (b), explain how a linear regression on transformed data could confirm the model's parameters.

[2]

Find the exact total amount of chemical using integration.

[3]

Calculate the absolute error of the trapezoidal estimate.

[2]

Suggest one way to improve the trapezoidal rule estimate's accuracy.

[1]

Question 6

SLPaper 2

A cylindrical container's radius, $r(h)$ , in centimeters, varies with height $h$ , in centimeters, from the base. The volume of liquid it can hold is given by $\pi \int_{0}^{5} r(h)^{2} d h$ . The radii are measured at 1 cm intervals:

$h(\mathrm{~cm})$	0	1	2	3	4
5
$r(h)(\mathrm{cm})$ 2	3	4	5	4	3

Use the trapezoidal rule with 5 intervals to estimate the volume of the container.

[4]

Sketch the graph of $r(h)^{2}$ against $h$ and shade the region used in part (a).

[2]

Question 7

SLPaper 2

A designer models the height of an arch, $h(x)$ , in meters, over a 10-meter span, where $x$ is the horizontal distance from the left support. The arch's height is given by $h(x)= -0.04 x^{2}+0.4 x$ , for $0 \leq x \leq 10$ . The area under the arch represents the space available for passage.

Use the trapezoidal rule with 5 intervals to estimate the area under the arch.

[3]

Sketch the graph of $h(x)$ and shade the region corresponding to the estimate in part (a).

[2]

Find the exact area under the arch using integration.

[3]

Question 8

SLPaper 2

A transportation company is studying the velocity of a delivery vehicle over time, modeled by the function $v(t) = 4t - t^2$ in meters per second, from $t=0$ to $t=4$ seconds.

Divide the interval $[0,4]$ into 4 equal sub-intervals. Write the values of $t$ at each endpoint.

[2]

Calculate the values of the function $v(t) = 4t - t^2$ at each endpoint.

[2]

Use the trapezoidal rule with these values to approximate the total distance travelled by the particle.

[3]

Verify the result by calculating the exact integral of $v(t)$ over the interval $[0,4]$ .

[4]

Compare the approximation with the exact answer and state which is larger. Explain why the difference might occur.

[3]

Question 9

SLPaper 2

A renewable energy company is analyzing the growth of a new type of solar panel efficiency over time. The efficiency of the solar panel is modeled by the function $f(x) = e^x$ over the interval $0 ≤ x ≤ 2$ years.

Divide the interval $[0,2]$ into 4 equal sub-intervals and write down the values of $x$ at each endpoint.

[3]

Use the trapezoidal rule with 4 sub-intervals to approximate the area under the curve $f(x) = e^x$ from $x=0$ to $x=2$ .

[7]

Use your calculator to find the exact value of the integral and compare it with the approximation from part (b).

[10]

Question 10

SLPaper 2

A city is planning to build a new park, and the area of the park can be modeled by the function $f(x) = x^2$ , where $x$ represents the length of one side of the park in meters, defined on the interval $0 \leq x \leq 2$ .

Divide the interval $[0, 2]$ into 4 equal sub-intervals and write down the $x$ -values.

[3]

Evaluate $f(x) = x^2$ at each endpoint.

[4]

Use the trapezoidal rule to approximate the area under the curve of the park's area function.

[5]

Verify the exact area by integrating $f(x)$ from 0 to 2.

[5]

Compare the approximation with the exact value.

[3]

x(\mathrm{~m})

h(x)(\mathrm{m})

0.88

1.92

2.52

2.56

2.00

h(\mathrm{~m})

A(h)\left(\mathrm{m}^{2}\right)

t(\mathrm{~m})

s(t)(\mathrm{m})

2.5

t(\mathrm{~h})

C(t)(\mathrm{mg} / \mathrm{L})

9.07

2.70

3.64

4.95

6.70

h(\mathrm{~cm})

r(h)(\mathrm{cm})