Volumes of Revolution (DP IB Maths: AI HL)

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Author

Paul

Last updated

16 July 2024

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Volumes of Revolution Around x-axis

What is a volume of revolution around the x-axis?

A solid of revolution is formed when an area bounded by a function $y = f (x)$
(and other boundary equations) is rotated $2 π$ radians $(360 °)$ around the $x$ -axis
The volume of revolution is the volume of this solid

2An illustration showing how a volume of revolution about the x-axis is formed from an area under a curve

Be careful – the ’front’ and ‘back’ of this solid are flat
- they were created from straight (vertical) lines
- 3D sketches can be misleading

How do I solve problems involving the volume of revolution around the x-axis?

Use the formula

$V = π \int_{a}^{b} y^{2} d x$

- This is given in the formula booklet
- $y$ is a function of $x$
- $x = a$ and $x = b$ are the equations of the (vertical) lines bounding the area
  - If $x = a$ and $x = b$ are not stated in a question, the boundaries could involve the $y$ -axis ( $x = 0$ ) and/or a root of $y = f (x)$
  - Use a GDC to plot the curve, sketch it and highlight the area to help
Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help

STEP 1

If a diagram is not given, use a GDC to draw the graph of

y = f (x)

If not identifiable from the question, use the graph to find the limits $a$ and $b$

STEP 2

Use a GDC and the formula to evaluate the integral

Thus find the volume of revolution

Examiner Tip

Functions involved can be quite complicated so type them into your GDC carefully
Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems

Worked example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of $y = \sqrt{3 x^{2} + 2}$ , the coordinate axes and the line $x = 3$ by $2 π$ radians around the $x$ -axis. Give your answer as an exact multiple of $π$ .

5-4-4-ib-hl-ai-aa-extraaa-ai-we1-soltn

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Volumes of Revolution Around y-axis

What is a volume of revolution around the y-axis?

Very similar to above, this is a solid of revolution which is formed when an area bounded by a function $y = f (x)$ (and other boundary equations) is rotated $2 π$ radians $(360 °)$ around the $y$ -axis
The volume of revolution is the volume of this solid

How do I solve problems involving the volume of revolution around y-axis?

Use the formula

$V = π \int_{a}^{b} x^{2} d y$

- This is given in the formula booklet
- $x$ is a function of $y$
  - the function is usually given in the form $y = f (x)$
  - this will need rearranging into the form $x = g (y)$
- $y = a$ and $y = b$ are the equations of the (horizontal) lines bounding the area
  - If $y = a$ and $y = b$ are not stated in the question, the boundaries could involve the $x$ -axis ( $y = 0$ ) and/or a root of $x = g (y)$
  - Use a GDC to plot the curve, sketch it and highlight the area to help
Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help

STEP 1

If a diagram is not given, use a GDC to draw the graph of

y = f (x)

(or

x = g (y)

if already in that form)

If not identifiable from the question use the graph to find the limits $a$ and $b$

STEP 2

If needed, rearrange

y = f (x)

into the form

x = g (y)

STEP 3

Use a GDC and the formula to evaluate the integral

A GDC will likely require the function written with '

x

' as the variable (not '

y

Thus find the volume of revolution

Examiner Tip

Functions involved can be quite complicated so type them into your GDC carefully
Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems

Worked example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of $y = x^{3} + 8$ and the coordinate axes by $2 π$ radians around the $y$ -axis. Give your answer to three significant figures.

5-4-4-ib-hl-ai-aa-extraaa-ai-we2-vor-soltn