IB Maths AI SL AI 5.2 Integration and kinematics questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Easy6 marks

A surveyor estimates the area of an irregular lake by measuring its width at equal intervals along a straight line through the lake. The measured widths are shown below.

Distance along the line, $x$ (m)	0	25	50	75	100
Width of lake (m)	0	18	24	17	0

(a)

Using the trapezoidal rule, show that the estimated area of the lake between $x=0$ and $x=50$ is $750\text{ m}^2$

[3]

(b)

Using your answer to part (a) and the remaining measurements, calculate an estimate of the total area of the lake.

[3]

Question 2

Paper 2Hard8 marks

During a test, a small cart moves along a straight track. For $0\leq t\leq 8$ its acceleration is modelled by

\frac{dv}{dt}=-0.072t^2+0.610t-1.05\quad\text{m s}^{-2},

where $t$ is in seconds and $v$ is in $\text{m s}^{-1}$ The cart’s velocity at $t=0$ is $1.02\text{ m s}^{-1}$

(a)

Show that

v(t)=-0.024t^3+0.305t^2-1.05t+1.02.

[2]

(b)

Use your GDC to find all values of $t$ for which the cart is momentarily at rest. Give your answers to three significant figures.

[2]

(c)

A technician claims that, since the cart starts moving in the positive direction, its forward distance over $0\leq t\leq 8$ is $\int_0^8 v(t)\,dt$

Assess this claim, giving a reason. Then determine the total distance travelled while the cart is moving in the positive direction. Use your results from part (b), write the definite-integral expression(s) before evaluating them with your GDC, and give the distance to three significant figures.

[4]

The best questions are still waiting

Unlock every question in this topic, then type, photograph or draw your answer and have it marked against the markscheme. Find out where you are dropping marks.

Unlock all questions Compare Student and Student Plus