5.11 Further integration questions for IB Maths AI

Find ${\displaystyle \int f(x)\mathrm{d}x}$.

Hence find the exact value of ${\displaystyle {\int }_1^{4}f(x)\mathrm{d}x}$, giving your answer in the form $a+b\ln 2$, where $a,b\in \mathbb{Z}$.

Write down the rate at which oil is leaking when the crack is found.

Find the volume of oil that leaks in the first 10 hours.

Let $L(t)$ be the total volume of oil, in litres, that has leaked $t$ hours after the crack is found. By integrating $r(t)$, show that $L(t)=10t+250-250e^{-0.2t}$.

Find the time taken for 400 litres of oil to leak from the tank.

Find the time at which the tank becomes empty, assuming the leak is not repaired.

Describe what happens to the rate of leaking as $t$ increases, and state one reason why the model may be unrealistic for large values of $t$.

Find the equation of the regression line of $P$ on $v$.

Write down the coefficient of determination, $R^2$, for this linear model.

Calculate the residuals of the linear model at $v=4$, $v=8$ and $v=12$, and hence explain why a linear model is not appropriate, despite the high value of $R^2$.

Tomasz next considers a power model $P=a{v}^b$.

Show that this model gives a linear relationship between $\ln P$ and $\ln v$.

Use your graphic display calculator to find the values of $a$ and $b$.

For a model $P=f(v)$, the sum of square residuals is $S{S}_{res}={\sum }_{i=1}^9(P_i-f(v_i){)}^2$.

Find the value of $S{S}_{res}$ for the power model.

Physical theory suggests that the power output is proportional to the cube of the wind speed, so Tomasz now considers the model $P=k{v}^3$. He chooses $k$ to minimize

Show that $S(k)$ is minimized when $k={\displaystyle \frac{\sum v_i^3{P}_i}{\sum v_i^6}}$.

Hence find the value of $k$.

For the model $P=k{v}^3$, $S{S}_{res}=516$, correct to three significant figures.

Tomasz notices that this one-parameter model has a smaller value of $S{S}_{res}$ than the power model in part (b), which has two parameters. Explain why this is possible, and state which of the two models fits the data better.

On one day, the wind speed at the site $t$ hours after midnight is modelled by $v=7+3\sin \left(\frac{\pi t}{12}\right)$, for $0\le t\le 24$.

The energy, in kWh, produced by the turbine in one day is ${\int }_0^{24}P\mathrm{d}t$. Use the model $P=k{v}^3$ to find the energy produced that day.

The turbine's maximum power output is 1500 kW. Comment on using the model $P=k{v}^3$ to predict the power output when the wind speed is 15 m s⁻¹.

Find ${\displaystyle \int g(x)\mathrm{d}x}$.

The region $R$ is enclosed by the graph of $g$, the $x$-axis and the lines $x=0$ and $x=2$. Find the exact area of $R$, giving your answer in the form $a\ln b$, where $a,b\in {\mathbb{Z}}^{+}$.

Write down the maximum rate at which water flows into the reservoir.

Find the exact volume of water that flows into the reservoir in the first 6 hours.

Find the exact volume of water that is pumped out of the reservoir in the first 6 hours.

Hence find the volume of water in the reservoir at 06:00.

Show that the volume of water in the reservoir, $V{\text{ m}}^3$, at time $t$ is given by

Find the maximum volume of water in the reservoir during the 36 hours, and the time at which it occurs.

Find the time at which the volume of water in the reservoir first falls below $2000{\text{ m}}^3$.

Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.

The tram company's engineer, Mireille, models the velocity by

Find the distance between the stops according to this model.

Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.

Use differentiation to find the maximum velocity of the tram according to Mireille's model.

Find the maximum acceleration of the tram according to the model, and the time at which it occurs.

For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².

Show that Mireille's model satisfies this condition.

Find the time at which the tram has travelled half of the distance between the stops.

On a new route, two stops are 900 m apart. Mireille uses a model of the same form,

where $T$ seconds is the journey time and $c$ is a positive constant.

Show that $c={\displaystyle \frac{10800}{T^4}}$.

Show that the greatest magnitude of the acceleration during the journey is ${\displaystyle \frac{10800}{T^2}}$.

Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).

Find the maximum velocity of the tram on this journey.

State one reason why Mireille's model may not be suitable for the new route.