IBMathsDPApplications & Interpretation (AI)HLExam Questions5. CalculusIntegration

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

4 hours30 questions
Easy
Medium
Hard
Very Hard
1a
2 marks

A function f is defined by the equation  f left parenthesis x right parenthesis equals negative 3 x plus 35.

Sketch the graph of  y equals f left parenthesis x right parenthesis in the interval  0 less or equal than x less or equal than 10.

How did you do?

1b
3 marks

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

\int_{1}^{9} \left(-3x + 35\right) \mathrm{d}x

You should not use your GDC to find the value of the integral.

How did you do?

1a
2 marks

Let f'\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}.

How did you do?

1b
2 marks

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

How did you do?

1c
4 marks

Find f\left(x\right).

How did you do?

2a
2 marks

Let f'\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}.

How did you do?

2b
2 marks

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

How did you do?

2c
4 marks

Find f\left(x\right).

How did you do?

3a
2 marks

The following table shows the x and space y coordinates of five points that lie on a curve y equals f left parenthesis x right parenthesis.

x

0

0.25

0.5

0.75

1

y equals f left parenthesis x right parenthesis

1

2.25

4

6.25

9

Estimate the area under the curve over the interval  0 less or equal than x less or equal than 1.

How did you do?

3b
2 marks

The equation of the curve was found to be  y equals left parenthesis 2 x plus 1 right parenthesis squared.

Find the exact value of the area under the curve over the interval 0 less or equal than x less or equal than 1.

How did you do?

3c
2 marks

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

How did you do?

4a
4 marks

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.

ib4-ai-sl-5-2-ib-maths-medium

Find

(i) the coordinates of \mathrm{P}

(ii) the coordinates of \mathrm{V}

(iii) the value of a.

How did you do?

4b
2 marks

Find the cross-sectional area under the arch.

How did you do?

5a
2 marks

The diagram below shows a part of the curve  y equals negative 4 x squared plus p x plus q.  Points A and B represent the x-intercepts, point  Vleft parenthesis 2.5 comma 6 right parenthesis  represents the vertex of the curve, and the shaded region straight R represents the area between the curve and the x-axis.

ib5-ai-sl-5-2-ib-maths-medium

Find the values of p and q.

How did you do?

5b
4 marks

Find the coordinates of points straight A and straight B.

How did you do?

5c
2 marks

Find the area of region straight R.

How did you do?

6a
2 marks

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

Graph of y = f(x), a parabola opening downwards that crosses the negative x-axis, the positive y-axis and the positive x-axis. The region R, bounded by the y-axis, the positive x-axis and the curve, is shaded.

Write down an integral for the area of R.

How did you do?

6b
1 mark

Find the area of R.

How did you do?

6c
2 marks

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.

Triangle ABC on a pair of coordinate axes. A is at the origin, C is on the positive x-axis, and B is above the x-axis, between A and C.

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

Find the value of h.

How did you do?

7a
3 marks

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 \geq 0.

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

How did you do?

7b
5 marks

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

Find P\left(x\right).

How did you do?

8a
3 marks

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 \geq 0.

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

How did you do?

8b
5 marks

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

Find P\left(x\right).

How did you do?

9a
3 marks

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 equals straight q left parenthesis straight x minus 6 right parenthesis squared .  Point straight O is the origin, and pointsstraight O comma straight A comma straight B and straight C  are the vertices of a rectangle.  Point straight V, the deepest point of the riverbed, is situated on the x-axis.

ib7-ai-sl-5-2-ib-maths-medium

Find

(i) the coordinates of straight V

(ii) the area of the rectangle OABC.

How did you do?

9b
2 marks

Determine the value of straight q.

How did you do?

9c
3 marks

Find the cross-sectional area of the riverbed.

How did you do?

10a
2 marks

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.

ib10-ai-sl-5-2-ib-maths-medium

Find the value of r.

How did you do?

10b
4 marks

Find the cross-sectional area of the trough.

How did you do?

10c
2 marks

The trough is 1.2 m long.

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

How did you do?

11a
2 marks

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.

ib5-ai-sl-5-2-ib-maths-hard

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.

How did you do?

11b
4 marks

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.

How did you do?

12a
3 marks

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

ib4-ai-sl-5-2-ib-maths-hard

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.

How did you do?

12b
3 marks

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.

How did you do?

13a
2 marks

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 \geq 0

Sketch the graph of v for 0 \leq t \leq 5.

How did you do?

13b
2 marks

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.

How did you do?

14
6 marks

Let

f'\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).

How did you do?

15a
3 marks

A function f is a piecewise linear function defined by

f\left(x\right) = \left\{\begin{matrix} \frac{1}{2}\left(x + 7\right), & x \leq 3 \\ 5, & 3 < x < 10 \\ 35 - 3x, & x \geq 10 \end{matrix}\right.

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

How did you do?

15b
4 marks

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

integral subscript 1 superscript 11 f left parenthesis x right parenthesis d x

How did you do?

16a
5 marks

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

ib3-ai-sl-5-2-ib-maths-veryhard

Find

(i) the value of a

(ii) f\left(x\right)

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

How did you do?

16b
3 marks

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

How did you do?

1a
5 marks

Let f'\left(x\right) = 4ax + 6, where a \in \mathbb{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}.

ib3-ai-sl-5-2-ib-maths-hard

Find

(i) the value of a

(ii) f\left(x\right)

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

How did you do?

1b
3 marks

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

How did you do?

2a
3 marks

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 \mathbb{R}. The shaded region R is bounded by the x-axis, the y-axis and the graph of f.

ib6-ai-sl-5-2-ib-maths-hard

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

(ii) Find the area of R.

How did you do?

2b
3 marks

\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.

ib6b-ai-sl-5-2-ib-maths-hard

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

How did you do?

3a
3 marks

A curve has the equation y equals negative x cubed plus 8 x squared minus 13 x plus 6 .  Consider the area enclosed by the curve and the positive x-axis.

Sketch the curve, shading the area indicated above.

How did you do?

3b
5 marks

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

How did you do?

3c
2 marks

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.

How did you do?

3d
3 marks

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

How did you do?

3e
2 marks

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

How did you do?

4a
3 marks

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.

ib8-ai-sl-5-2-ib-maths-hard

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.

How did you do?

4b
5 marks

Find the area of R.

How did you do?

5a
6 marks

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 \leq x \leq 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).

How did you do?

5b
7 marks

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.

How did you do?

6
6 marks

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:

ib4-ai-sl-5-2-ib-maths-veryhard

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.

How did you do?

7
6 marks

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:

ib5-ai-sl-5-2-ib-maths-veryhard

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.

How did you do?

8
7 marks

Let

f'\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).

How did you do?

9a
4 marks

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 \leq t \leq 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.

How did you do?

9b
5 marks

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}.

How did you do?

1a
3 marks

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 \mathbb{R}. The shaded region R is bounded by the x-axis, the y-axis and the graph of f.

ib6-ai-sl-5-2-ib-maths-veryhard

Find the area of region R.

How did you do?

1b
6 marks

A trapezoid \mathrm{ABCD} is shown below.

ib6a-ai-sl-5-2-ib-maths-veryhard

\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}.

How did you do?

2a
2 marks

A curve has the equation  y equals x cubed minus 5 x squared plus 2 x plus 8.

Sketch the curve.

How did you do?

2b
5 marks

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

integral subscript negative 1 end subscript superscript 3 over 2 end superscript left parenthesis x cubed minus 5 x squared plus 2 x plus 8 right parenthesis d x

Give your answer as an exact value.

How did you do?

2c
3 marks

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

How did you do?

2d
3 marks

(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.

How did you do?

2e
1 mark

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

How did you do?

3a
4 marks

The shaded region  in the following diagram is bounded by the two curves     y equals 5 x squared minus 12 x plus 8 and  y equals negative 3 x squared plus 10 x plus 29 .

ib8-ai-sl-5-2-ib-maths-veryhard

The two curves intersect at points straight A and straight B as shown.  x subscript A and x subscript B are the x-coordinates of points straight A and straight B respectively.

By setting up and solving an appropriate quadratic equation, find the values of x subscript A and x subscript B

How did you do?

3b
6 marks

Find the area of region straight R , giving your answer as an exact value.

How did you do?

4a
5 marks

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 \leq x \leq 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).

How did you do?

4b
6 marks

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.

How did you do?

4c
4 marks

(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.

How did you do?