DP IB Maths: AI SL

Topic Questions

5.2 Integration

1a
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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.

1b
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3 marks

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

integral subscript 1 superscript 9 left parenthesis negative 3 x plus 35 right parenthesis straight d x

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

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2
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6 marks

The derivative of the function f is given by

f to the power of apostrophe left parenthesis x right parenthesis equals 9 over 2 x squared plus 7 x minus 2

and the curve y equals f left parenthesis x right parenthesis   passes through the point open parentheses negative 3 comma negative 11 over 2 close parentheses .

Find an expression for f.

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3a
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5 marks

A curve y equals f left parenthesis x right parenthesis  has the gradient function f to the power of apostrophe left parenthesis x right parenthesis equals 4 a x plus 6  ,  where a element of straight real numbers is a constant.  The diagram below shows part of the curve, with the x and y intercepts labelled and where straight V represents the vertex of the curve.

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

Find

(i)

the value of a

(ii)

the equation of the curve y equals f left parenthesis x right parenthesis

(iii)
the coordinates of straight V.
3b
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3 marks

Find the area between the curve and the x-axis.

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4a
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3 marks

A section of the curve with equation   y equals 1 half open parentheses x minus 1 close parentheses open parentheses x plus 5 close parentheses  is shown below:

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

The shaded region S in the diagram is bounded by the curve, the x-axis and the line  x equals 2.

(i)
Write down an integral for the area of the shaded region straight S.

(ii)
Find the area of straight S.  Give your answer as a fraction.
4b
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3 marks

The shaded region straight R in the diagram is bounded on three sides by the curve, the x-axis and the y-axis.  The boundary on the fourth side is a straight line parallel to the x-axis, and that line, the curve and the line  x equals 2  all intersect at a single point.

Find the area of region straight R. Give your answer as a fraction.

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5a
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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 a cross-sectional area as indicated by the shaded region straight R in the following diagram:

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

Region straight R is bounded, as shown, by the positive x- and y-axes and the curve with equation y equals fraction numerator 6 open parentheses x minus 3 close parentheses over denominator 2 x minus 9 end fraction .  All units are in centimetres.

Using technology, or otherwise, find the coordinates of the points of intersection of the curve with the x- and y-axes.

5b
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4 marks

The volume of the puzzle piece is to be 30 cm cubed.

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

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6a
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3 marks

The following diagram shows part of the graph of  f left parenthesis x right parenthesis equals left parenthesis 2 x plus 1 right parenthesis left parenthesis 4 x squared minus 10 x plus 41 right parenthesisx element of straight real numbers .  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 region straight R.

(ii)
Find the area of region straight R.
6b
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3 marks

ABCD is a parallelogram with vertices straight A left parenthesis 0 comma 0 right parenthesis, straight B open parentheses 1 comma 7 over 3 close parenthesesstraight C and straight D left parenthesis a comma 0 right parenthesis, as shown in the diagram below.  The area of ABCD is equal to the area of region straight R above.

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

By first finding the value of a, the x-coordinate of point straight D, determine the coordinates of point straight C. The coordinates should be given as exact fractions.

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7a
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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.

7b
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5 marks

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

7c
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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.

7d
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3 marks

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

7e
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2 marks

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

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8a
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3 marks

The shaded region straight R in the following diagram is bounded by the x-axis, the line  y equals 8 x minus 4  and the curve  y equals negative x cubed plus x squared plus 10 x plus 8.

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

Using technology, or otherwise, find the coordinates of

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

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

(iii)
the point of intersection between the curve and the x-axis that is shown in the diagram.
8b
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6 marks

Show that the area of region straight R is equal to exactly 439 over 12 units2 .  Be sure to show all of your working.

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9a
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2 marks

For a particle straight 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 parenthesis t right parenthesis equals 2 t squared minus 8 t plus 9 comma space space space space space space space t greater or equal than 0

Sketch the graph of v left parenthesis t right parenthesis in the interval 0 less or equal than t less or equal than 5 .

9b
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3 marks

The distance travelled between times t subscript 1 and t subscript 2 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 the distance travelled by the particle straight P between the times   t equals 1 and  t equals 4.5.

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10a
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6 marks

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

fraction numerator dM over denominator straight d x end fraction equals negative 0.068 x cubed plus 0.72 x squared minus 0.88 x minus 1.9 comma space space 0 less or equal than x less or equal than 10

where straight M represents the number of meals sold in a week (in thousands of meals sold), and x represents the amount spent on advertising during the preceding week (in thousands of euros).

It is known as well that 5988 meals are sold in a week where 2000 euros had been spent on advertising during the preceding week.

Find an expression for straight M left parenthesis x right parenthesis.

10b
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7 marks

Find the maximum number of meals that the company can expect to sell in a week, and the amount of money that the company should spend on advertising during the preceding week to bring about that level of sales. Give your answers to the nearest meal sold and the nearest euro, respectively.  Be sure to justify that the value you find is indeed a maximum.

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1a
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2 marks

A curve y equals f left parenthesis x right parenthesis passes through point  straight A left parenthesis 4 comma 2 right parenthesis and has a gradient of f to the power of apostrophe left parenthesis x right parenthesis equals 5 x minus 2 .

Find the gradient of the curve at point straight A.

1b
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2 marks

Find the equation of the tangent to the curve at point straight A.

Give your answer in the form  y equals m x plus c.

1c
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3 marks

Determine the equation of the curve  y equals f left parenthesis x right parenthesis.

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2a
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2 marks

A point  P left parenthesis 3 comma 8 right parenthesis  lies on the curve y equals f left parenthesis x right parenthesis that has a gradient of  f to the power of apostrophe left parenthesis x right parenthesis equals negative 2 x squared plus 11.

Find the gradient of the curve at point straight P.

2b
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2 marks

Find the equation of the tangent to the curve at point straight P.

Give your answer in the form y equals m x plus c .

2c
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3 marks

Determine the equation of the curve y equals f left parenthesis x right parenthesis.

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3a
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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.

3b
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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.

3c
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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.

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4a
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4 marks

The following diagram shows an arch that is 4.5 space straight m tall and 3 space straight m wide. The arch crosses the x-axis at the origin, straight O, and at point straight P, and its vertex is at point straight V.  The arch may be represented by a curve with an equation of the form y equals x left parenthesis a x plus 6 right parenthesis ,  where all units are measured in metres.

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

Find

(i)

the coordinates of straight P

(ii)

the coordinates of straight V

(iii)
the value of a.
4b
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2 marks

Find the cross-sectional area under the arch.

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5a
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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.

5b
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4 marks

Find the coordinates of points straight A and straight B.

5c
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2 marks

Find the area of region straight R.

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6a
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2 marks

The following diagram shows part of the graph of f left parenthesis x right parenthesis equals left parenthesis 5 minus 2 x right parenthesis left parenthesis 2 plus 3 x right parenthesis comma space space x element of R .  The shaded region straight R is bounded by the x-axis, the y-axis and the graph of f.

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

Write down an integral for the area of region straight R

6b
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1 mark

Find the area of region straight R.

6c
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2 marks

The three points  straight A left parenthesis 0 comma 0 right parenthesis comma space straight B left parenthesis 4 comma h right parenthesis  and  straight C left parenthesis 9 comma 0 right parenthesis define the vertices of a triangle.

q5-3-integration-ib-aa-sl

Find the value of h, the y-coordinate of straight B, given that the area of the triangle is equal to the area of region straight R.

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7a
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3 marks

A rice farm sells x kg of rice every week.

It is known that  fraction numerator straight d P over denominator straight d x end fraction equals negative 0.02 x plus 6 comma space x greater or equal than 0 comma  where P is the weekly profit, in dollars ($), from the sale of x kg of rice.

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

7b
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5 marks

The profit from selling 250 spacekg of rice is $480.

Find P left parenthesis x right parenthesis.

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8a
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3 marks

A paint company sells x hundred of litres of paint every week.

It is known that  fraction numerator dP over denominator straight d x end fraction equals negative 1.9 x plus 145 comma space x greater or equal than 0 comma  where straight P is the weekly profit, in euros (€), from the sale of x hundred litres of paint.

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

8b
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5 marks

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

Find P left parenthesis x right parenthesis.

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9a
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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.
9b
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2 marks

Determine the value of straight q.

9c
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3 marks

Find the cross-sectional area of the riverbed.

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10a
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2 marks

A trough has a cross-sectional area shown by the shaded region of the diagram below, where the x and y values are in centimetres.  The curved bottom of the trough has an equation in the form y equals r left parenthesis x minus 15 right parenthesis squared .  Point straight O is the origin, and points  straight O comma straight A comma straight B space and space straight C are the vertices of a rectangle.  Point straight V, the deepest point of the trough, is situated on the x-axis.

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

Determine the value of r.

10b
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4 marks

Find the cross-sectional area of the trough.

10c
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2 marks

The length of the trough is 1.2 m.

Find the volume of the trough. Give your answer in cm3

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1a
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3 marks

A function f is a piecewise linear function defined by

f open parentheses x close parentheses equals open curly brackets table row cell 1 half open parentheses x plus 7 close parentheses comma space space space space space space space space space space space space space space x less or equal than 3
space space space space space 5 comma space space space space space space space space space space space space space space space space space space 3 less than x less than 10
space space 35 minus 3 x comma space space space space space space space space space space space space space space space x greater or equal than 10 end cell blank row blank blank end table close

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 12 .

1b
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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

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2
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7 marks

The derivative of the function f is given by

f to the power of apostrophe left parenthesis x right parenthesis equals negative 3 over x squared plus 1 half x squared minus 2 over 3 x plus 2 comma space space space x greater than 0

and the curve  y equals f left parenthesis x right parenthesis  passes through the point open parentheses 6 comma   65 over 2 close parentheses .

Find an expression for f.

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3a
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5 marks

A curve  y equals f left parenthesis x right parenthesis  has the gradient function  f to the power of apostrophe left parenthesis x right parenthesis equals a x minus 1 .  The diagram below shows part of the curve, with the x- and y-intercepts labelled.

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

Find

(i)

the value of a

(ii)

the equation of the curve y equals f left parenthesis x right parenthesis

(iii)
the coordinates of point straight M.
3b
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3 marks

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

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4
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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 equals 10  and y equals 6 .  The shape of one of the curved sections in the diagram can be described by the curve with equation

y equals negative 1 fourth open parentheses 2 x minus 7 close parentheses open parentheses 2 x minus 13 close parentheses

All units are given in centimetres.

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

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5
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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 straight R in the following diagram:

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

Region straight R is bounded, as shown, by the positive x-axis and the curve with equation  y equals space. All units are in centimetres.

Given that the model piece will have a volume of 50.3 cm3, find the length of the piece.

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6a
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3 marks

The following diagram shows part of the graph of f left parenthesis x right parenthesis equals 1 half open parentheses 2 x plus 1 close parentheses open parentheses x squared minus 5 over 2 x plus 8 close parentheses comma space space space x element of straight real numbers, .  The shaded region straight 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  straight R

6b
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6 marks

A trapezoid ABCD is shown below.

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

left square bracket AB right square bracket is perpendicular to left square bracket AD right square bracket and parallel to left square bracket CD right square bracketCD equals 34 over 45  .  The coordinates of points straight Astraight B and straight D are left parenthesis 0 comma 1 right parenthesis, open parentheses 9 over 10 comma p close parentheses andopen parentheses 3 over 4 comma 7 over 16 close parentheses  respectively, where p greater than 0 is a constant.

 Given that ABCD has the same area as the region R above, find the value of p, the   y-coordinate of point straight B.

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7a
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2 marks

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

Sketch the curve.

7b
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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.

7c
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3 marks

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

7d
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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.
7e
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1 mark

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

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8a
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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

8b
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6 marks

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

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9a
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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 parenthesis t right parenthesis equals t cubed minus 15 t squared plus 48 t plus 64 comma space space space space space space 0 less or equal than t less or equal than 10

At time t subscript 1 the particle reaches its maximum velocity, while at time t subscript 2 the particle comes momentarily to rest.

Find the values of t subscript 1 spaceand t subscript 2, justifying your answers in each case.

9b
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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 straight P between times   t equals 0  and  t equals 10 .

(ii)
the percentage of that total distance that is covered between times t subscript 1 and t subscript 2.

 

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10a
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11 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, and after analysing several years of data she determines that the rate of change of his number of ‘likes’ can be modelled by the equation

fraction numerator dL over denominator straight d x end fraction equals negative 0.164 x cubed plus 2.73 x squared minus 12.7 x plus 15.3 comma space space space 0 less or equal than x less or equal than 12

where straight L represents the number of likes received on a given day (in thousands of likes), and x represents the amount of new video content Donty uploaded on the preceding day (in hours).  Because of technical limitations, Donty is unable to upload more than 12 hours of new video content on any given day.

It is known as well that 36075 ‘likes’ are received on a day after 5 hours of video content was uploaded the day before

Find the maximum and minimum number of ‘likes’ that Donty can expect to receive in a day, and the corresponding number of hours of new video content that Donty should upload on the preceding day to attain that maximum or minimum. Be sure to justify that the values you find are indeed the maximum and minimum possible.

10b
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4 marks
(i)
For the maximum value determined in part (a), calculate the number of likes that are received for each minute of new video content uploaded the preceding day.

(ii)
State, with a reason, whether the value calculated in part (b) (i) represents the maximum number of ‘likes per minute of new content’ that Donty is able to achieve.

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