Integration (DP IB Applications & Interpretation (AI): SL): Exam Questions

Let $f\text{'}\left(x\right)=5x-2$. The graph of $f$ passes through the point $\mathrm{A}\left(4,2\right)$.

Find the gradient of the graph of $f$ at $\mathrm{A}$.

Find the equation of the tangent to the graph of $f$ at $\mathrm{A}$. Give your answer in the form $y=mx+c$.

Find $f\left(x\right)$.

Let $f\text{'}\left(x\right)=-2x^2+11$. The graph of $f$ passes through the point $\mathrm{P}\left(3,8\right)$.

Find the gradient of the graph of $f$ at $\mathrm{P}$.

Find the equation of the tangent to the graph of $f$ at $\mathrm{P}$. Give your answer in the form $y=mx+c$.

Find $f\left(x\right)$.

The following table shows the $x$ and $y$ coordinates of five points that lie on a curve $y=f\left(x\right)$.

Estimate the area under the curve over the interval  $0\le x\le 1$.

The equation of the curve was found to be  $y=(2x+1{)}^2$.

Find the exact value of the area under the curve over the interval $0\le x\le 1$.

Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.

The following diagram shows an arch that is 4.5 m tall and 3 m wide. The arch meets the $x$-axis at the origin, $\mathrm{O}$, and at the point $\mathrm{P}$, and its vertex is at the point $\mathrm{V}$. The arch can be modelled by the curve with equation $y=x\left(ax+6\right)$, where $x$ and $y$ are measured in metres.

Find

(i) the coordinates of $\mathrm{P}$

(ii) the coordinates of $\mathrm{V}$

(iii) the value of $a$.

Find the cross-sectional area under the arch.

The diagram below shows a part of the curve  $y=-4x^2+px+q$.  Points $A$ and $B$ represent the $x$-intercepts, point  V$(2.5,6)$  represents the vertex of the curve, and the shaded region $\mathrm{R}$ represents the area between the curve and the $x$-axis.

Find the values of $p$ and $q$.

Find the coordinates of points $\mathrm{A}$ and $\mathrm{B}$.

Find the area of region $\mathrm{R}$.

The following diagram shows part of the graph of $f\left(x\right)=\left(5-2x\right)\left(2+3x\right),x\in \mathrm{ℝ}$. The shaded region $R$ is bounded by the $x$-axis, the $y$-axis and the graph of $f$.

Write down an integral for the area of $R$.

Find the area of $R$.

The three points $\mathrm{A}\left(0,0\right)$, $\mathrm{B}\left(4,h\right)$ and $\mathrm{C}\left(9,0\right)$ are the vertices of a triangle.

The area of the triangle is equal to the area of $R$.

Find the value of $h$.

A rice farm sells $x$ kg of rice each week. The weekly profit, $P$ dollars, from the sale of $x$ kg of rice satisfies

$\frac{\mathrm{d}P}{\mathrm{d}x}=-0.02x+6$, for $x\ge 0$.

Find the amount of rice, in kg, that should be sold each week to maximise the profit.

The profit from selling 250 kg of rice in a week is 480 dollars.

Find $P\left(x\right)$.

A paint company sells $x$ hundred litres of paint each week. The weekly profit, $P$ euros (€), from the sale of $x$ hundred litres of paint satisfies

$\frac{\mathrm{d}P}{\mathrm{d}x}=-1.9x+145$, for $x\ge 0$.

Find the number of litres of paint that should be sold each week to maximise the profit.

The profit from selling 7000 litres of paint in a week is €5000.

Find $P\left(x\right)$.

A river has a cross-sectional area shown by the shaded region of the diagram below, where the $x$ and $y$ values are in metres.  The riverbed (the curved part of the region shown) has an equation of the form $y=\mathrm{q}{(\mathrm{x}-6)}^2$ .  Point $\mathrm{O}$ is the origin, and points$\mathrm{O},\mathrm{A},\mathrm{B}$ and $\mathrm{C}$  are the vertices of a rectangle.  Point $\mathrm{V}$, the deepest point of the riverbed, is situated on the $x$-axis.

Find

(i) the coordinates of $\mathrm{V}$

(ii) the area of the rectangle $\mathrm{OABC}$.

Determine the value of $\mathrm{q}$.

Find the cross-sectional area of the riverbed.

The shaded region in the diagram below shows the cross-section of a trough, where $x$ and $y$ are measured in centimetres. The curved bottom of the trough can be modelled by an equation of the form $y=r{\left(x-15\right)}^2$. Point $\mathrm{O}$ is the origin, and points $\mathrm{O}$, $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are the vertices of a rectangle. Point $\mathrm{V}$, the lowest point of the trough, lies on the $x$-axis.

Find the value of $r$.

Find the cross-sectional area of the trough.

The trough is 1.2 m long.

Find the volume of the trough. Give your answer in ${\mathrm{cm}}^3$.

A company is designing a plastic piece for a new game. The piece is to be in the form of a prism, with cross-section the shaded region $R$ in the following diagram.

Region $R$ is bounded by the positive $x$-axis, the positive $y$-axis and the curve with equation $y=\frac{6\left(x-3\right)}{2x-9}$. All lengths are in centimetres.

Find the coordinates of the points where the curve meets the $x$-axis and the $y$-axis.

The volume of the piece is to be 30 ${\mathrm{cm}}^3$.

Find the length of the piece, giving your answer correct to 3 significant figures.

The following diagram shows part of the graph of $y=\frac{1}{2}\left(x-1\right)\left(x+5\right)$.

The shaded region $S$ is bounded by the graph, the $x$-axis and the line $x=2$.

(i) Write down an integral for the area of $S$.

(ii) Find the area of $S$. Give your answer as an exact value.

The shaded region $R$ is bounded by the graph, the $x$-axis, the $y$-axis and the horizontal line through the point on the graph where $x=2$.

Find the area of $R$. Give your answer as an exact value.

For a particle $P$ travelling in a straight line, the velocity, $v$ m/s, of the particle at time $t$ seconds is given by

$v\left(t\right)=2t^2-8t+9,t\ge 0$

Sketch the graph of $v$ for $0\le t\le 5$.

The distance travelled between two times by a particle moving in a straight line may be found by finding the area under its velocity-time graph between those two times.

Find the distance travelled by $P$ between $t=1$ and $t=4.5$.

Let

$f\text{'}\left(x\right)=\frac{9}{2}{x}^2+7x-2$

The graph of $f$ passes through the point $\left(-3,-\frac{11}{2}\right)$.

Find $f\left(x\right)$.

A function $f$ is a piecewise linear function defined by

$f\left(x\right)=\{\begin{array}{cc}\frac{1}{2}\left(x+7\right),& x\le 3\\ 5,& 3<x<10\\ 35-3x,& x\ge 10\end{array}$

Sketch the graph of $y=f\left(x\right)$ in the interval $0\le x\le 12$.

Use your sketch from part (a), along with relevant area formulae, to work out the value of the integral

${\int }_1^{11}f\left(x\right)dx$

Let $f\text{'}\left(x\right)=ax-1$, where $a\in \mathrm{ℝ}$ is a constant. The diagram below shows part of the graph of $f$, with its $x$- and $y$-intercepts labelled.

Find

(i) the value of $a$

(ii) $f\left(x\right)$

(iii) the coordinates of $\mathrm{M}$.

Find the area of the region enclosed by the graph of $f$ and the $x$-axis.

Let $f\text{'}\left(x\right)=4ax+6$, where $a\in \mathrm{ℝ}$ is a constant. The diagram below shows part of the graph of $f$. The graph crosses the $x$-axis at $x=-1$ and $x=4$, and its vertex is at $\mathrm{V}$.

Find

(i) the value of $a$

(ii) $f\left(x\right)$

(iii) the coordinates of $\mathrm{V}$.

Find the area of the region enclosed by the graph of $f$ and the $x$-axis.

The following diagram shows part of the graph of $f\left(x\right)=\left(2x+1\right)\left(4x^2-10x+41\right),x\in \mathrm{ℝ}$. The shaded region $R$ is bounded by the $x$-axis, the $y$-axis and the graph of $f$.

(i) Write down an integral for the area of $R$.

(ii) Find the area of $R$.

$\mathrm{ABCD}$ is a parallelogram with vertices $\mathrm{A}\left(0,0\right)$, $\mathrm{B}\left(1,\frac{7}{3}\right)$, $\mathrm{C}$ and $\mathrm{D}\left(a,0\right)$, as shown in the diagram below. The area of $\mathrm{ABCD}$ is equal to the area of $R$.

Find the value of $a$. Hence find the coordinates of $\mathrm{C}$. Give your answers as exact values.

A curve has the equation $y=-x^3+8x^2-13x+6$ .  Consider the area enclosed by the curve and the positive $x$-axis.

Sketch the curve, shading the area indicated above.

Using the trapezoidal rule with 5 strips, determine an approximation for the shaded area.

Explain, using your sketch from part (a), why it is not possible to determine immediately whether your approximation will be an underestimate or an overestimate.

Using integration, determine the exact value of the shaded area.

Find the percentage error of the approximation found in part (b), compared with the exact value.

The shaded region $R$ in the following diagram is bounded by the $x$-axis, the line $y=8x-4$ and the curve $y=-x^3+x^2+10x+8$.

Find the coordinates of

(i) the point of intersection of the curve and the line

(ii) the point of intersection of the line and the $x$-axis

(iii) the point of intersection of the curve and the $x$-axis that is shown in the diagram.

Find the area of $R$.

After analysing several years of data, a fast food company has found that the rate of change of its weekly sales can be modelled by

$\frac{\mathrm{d}M}{\mathrm{d}x}=-0.068x^3+0.72x^2-0.88x-1.9$, for $0\le x\le 10$,

where $M$ is the number of meals sold in a week, in thousands, and $x$ is the amount spent on advertising in the previous week, in thousands of euros.

In a week after 2000 euros was spent on advertising, 5988 meals were sold.

Find an expression for $M\left(x\right)$.

Find the maximum number of meals that the company can expect to sell in a week, and the amount that it should spend on advertising in the previous week to achieve this. Give your answers to the nearest meal and the nearest euro respectively. Justify that the value you find is a maximum.

Celebrity chef Pepper Bee has opened a new restaurant and is charging diners £630 for a piece of his signature ‘Croesus’ cake. The chef claims that the price reflects the high cost of the gold foil that is placed on top of each slice of cake, but a suspicious and disgruntled customer has decided to investigate this claim.

The shaded area in the diagram below shows the shape of the piece of gold foil that is placed on top of each slice of cake:

The shape is that of a rectangle, from which four identical curved sections have been removed. The rectangle is bounded by the positive $x$- and $y$-axes and the lines $x=10$ and $y=6$. The shape of one of the curved sections in the diagram can be described by the curve with equation

$y=-\frac{1}{4}\left(2x-7\right)\left(2x-13\right)$

All units are given in centimetres.

Given that gold foil costs £0.004788 per ${\mathrm{mm}}^2$, work out the cost of the gold foil on a piece of Pepper Bee’s Croesus cake. Give your answer to 2 decimal places.

A company is designing a piece for one of the plastic wargaming models they produce. The piece is to be in the form of a prism, with a cross-sectional area as indicated by the shaded region $R$ in the following diagram:

Region $R$ is bounded, as shown, by the positive $x$-axis and the curve with equation $y=\frac{1}{x-3}-x^2+6x-4$. All units are in centimetres.

Given that the model piece will have a volume of 50.3 ${\mathrm{cm}}^3$, find the length of the piece.

Let

$f\text{'}\left(x\right)=-\frac{3}{x^2}+\frac{1}{2}{x}^2-\frac{2}{3}x+2,x>0$

The graph of $f$ passes through the point $\left(6,\frac{65}{2}\right)$.

Find $f\left(x\right)$.

For a particle $P$ travelling in a straight line, the velocity, $v$ m/s, of the particle at time $t$ seconds is given by the equation

$v\left(t\right)=t^3-15t^2+48t+64,0\le t\le 10$

At time $t_1$ the particle reaches its maximum velocity, while at time $t_2$ the particle comes momentarily to rest.

Find the values of $t_1$ and $t_2$, justifying your answers in each case.

The distance travelled between two times by a particle moving in a straight line may be found by finding the area beneath the particle’s velocity-time graph between those two times.

Find

(i) the total distance travelled by the particle $P$ between times $t=0$ and $t=10$

(ii) the percentage of that total distance that is covered between times $t_1$ and $t_2$.

The following diagram shows part of the graph of $f\left(x\right)=\frac{1}{2}\left(2x+1\right)\left(x^2-\frac{5}{2}x+8\right),x\in \mathrm{ℝ}$. The shaded region $R$ is bounded by the $x$-axis, the $y$-axis and the graph of $f$.

Find the area of region $R$.

A trapezoid $\mathrm{ABCD}$ is shown below.

$\left[\mathrm{AB}\right]$ is perpendicular to $\left[\mathrm{AD}\right]$ and parallel to $\left[\mathrm{CD}\right]$, and $\mathrm{CD}=\frac{34}{45}$. The coordinates of the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ are $\left(0,1\right)$, $\left(\frac{9}{10},p\right)$ and $\left(\frac{3}{4},\frac{7}{16}\right)$ respectively, where $p>0$ is a constant.

Given that $\mathrm{ABCD}$ has the same area as the region $R$ above, find the value of $p$, the $y$-coordinate of point $\mathrm{B}$.

A curve has the equation  $y=x^3-5x^2+2x+8$.

Sketch the curve.

Using the trapezoidal rule with $n=5$, determine an approximation for the integral

${\int }_{-1}^{\frac{3}{2}}(x^3-5x^2+2x+8)dx$

Give your answer as an exact value.

Explain, using your sketch from part (a), why your approximation will be an underestimate.

(i) Find the exact value of the integral from part (b).

(ii) Find the percentage error of the approximation found in part (b), compared with the exact value of the integral.

Explain how you might modify your method in part (b) in order to get a more accurate approximation.

The shaded region  in the following diagram is bounded by the two curves     $y=5x^2-12x+8$ and  $y=-3x^2+10x+29$ .

The two curves intersect at points $\mathrm{A}$ and $\mathrm{B}$ as shown.  $x_A$ and $x_B$ are the $x$-coordinates of points $\mathrm{A}$ and $\mathrm{B}$ respectively.

By setting up and solving an appropriate quadratic equation, find the values of $x_A$ and $x_B$

Find the area of region $\mathrm{R}$ , giving your answer as an exact value.

Donty is a would-be social media celebrity who is obsessed with the number of likes his posts receive. He hires a statistician to study his social media accounts. After analysing several years of data, she finds that the rate of change of his number of likes can be modelled by

$\frac{\mathrm{d}L}{\mathrm{d}x}=-0.164x^3+2.73x^2-12.7x+15.3$, for $0\le x\le 12$,

where $L$ is the number of likes received on a given day, in thousands, and $x$ is the amount of new video content, in hours, that Donty uploaded on the previous day. Because of technical limitations, Donty cannot upload more than 12 hours of new video content in a day.

On a day after Donty uploaded 5 hours of new video content, he received 36 075 likes.

Find an expression for $L\left(x\right)$.

Find the maximum and the minimum number of likes that Donty can expect to receive in a day, and the corresponding number of hours of new video content that he should upload on the previous day. Justify that the values you find are the maximum and the minimum.

(i) For the maximum found in part (b), calculate the number of likes received for each minute of new video content uploaded on the previous day.

(ii) State, with a reason, whether the value found in part (c)(i) is the greatest number of likes per minute of new content that Donty can achieve.