DP IB Maths: AA SL

Topic Questions

5.3 Integration

1a
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2 marks

Show that

open parentheses 3 minus 2 x close parentheses squared equals 9 minus 12 x plus 4 x squared space

1b
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2 marks

Hence, or otherwise, work out

integral open parentheses 3 minus 2 x close parentheses squared space straight d x

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2
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4 marks

Given

integral subscript k superscript 5 left parenthesis 2 x minus 1 right parenthesis space straight d x equals 20

find the value of the positive constant k.

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3a
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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.

3b
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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.

3c
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3 marks

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

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4a
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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.

4b
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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 .

4c
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3 marks

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

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5a
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1 mark

The diagram below shows part of the graph ofspace y equals 4 minus open parentheses x minus 2 close parentheses squared.

q5-8-1-integration-easy-a-level-maths-pure-screenshot

Write down the values of x where y equals 0.

5b
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1 mark

Show that

4 minus open parentheses x minus 2 close parentheses squared equals 4 x minus x squared

5c
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2 marks

Evaluate

space integral subscript 0 superscript 4 open parentheses 4 x minus x squared close parentheses space straight d x

5d
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1 mark

Write down the area of the region labelled R.

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6a
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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.
6b
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2 marks

Find the cross-sectional area under the arch.

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7a
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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 Care 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.

7b
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4 marks

Find the cross-sectional area of the trough.

7c
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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 cm cubed.

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8a
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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 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-medium

Write down an integral for the area of region straight R

8b
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1 mark

Find the area of region R.

8c
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2 marks

The three points  straight A left parenthesis 0 comma 0 right parenthesis comma 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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