Adding and Subtracting Volumes

As with the area between a curve and a line or the area between 2 curves, a required volume may be created by two functions
- In this note we focus on volumes created by rotation around the x-axis but the same principles apply to rotation around the y-axis
- Make sure you are familiar with the methods in Volumes of Revolution
The volumes created here can be created from areas that do not have the x-axis as one its boundaries
- A cylinder is created by rotating a rectangle that borders the x-axis around the x-axis by 360°
- An annular prism (a cylinder with a whole through it – like a toilet roll) is created by rotating a rectangle that does not have a boundary with the x‑axis around the x-axis by 360°
A rectangle would be defined by two vertical and two horizontal lines

6-2-5-cie-fig1-vor-toilet-roll

- $x = a, x = b, y = c, y = d$
  - Where a, b, c & d are all positive and a < b and c < d
- The volume of revolution of this rectangle would be
  - $V = π \int_{a}^{b} d^{2} d x - π \int_{a}^{b} c^{2} d x$

When the area to be rotated around an axis has more than one function (and an axis) defining its boundary it can be trickier to tell whether to add or subtract volumes of revolution
- It will depend on
  - The nature of the functions and their points of intersection
  - Whether rotation is around the x-axis or the y-axis
Consider the region R, bounded by a curve, a line and the -axis, in the diagram below

6-2-5-cie-fig1-1-curve-line

If R is rotated around the $x$ -axis the solid of revolution formed will have a ‘hole’ in its centre

6-2-5-cie-fig1-2-curve-line-x-rotate-1

- Think in 2D and area
  - “region under the curve”
    SUBTRACT
    “region under the line”

If R is rotated around the $y$ -axis the solid of revolution formed will look a little like a spinning top – with a ‘dome top half’ and a ‘cone bottom half’

6-2-5-cie-fig1-3-curve-line-y-rotate

- Think in 2D and area
  - “top ‘half’ is the area ‘below’ the curve to the horizontal where the curve and line intersect”
    ADD
    “bottom ‘half’ is area ‘below’ the line to the horizontal where the curve and line interest”

Visualising the solid created becomes increasingly useful (but also trickier) for shapes generated by separate volumes of revolution
- Continue trying to sketch the functions and their solids of revolution to help

STEP 1: Identify the axis that the area will be rotated around
- Identify the functions $(y_{1}, y_{2}, . . .)$ involved in generating the volume
- Determine whether these will need to be added or subtracted

STEP 2: If rotating around the x-axis, square y for all functions
- If rotating around the y-axis, rearrange all the y functions into the form $x = g (y)$ and square
- In either case do this first without worrying about π or the integration and limits

STEP 3: Identify the limits for each volume involved and form the integrals required
- The limits could come from a graph

STEP 4: Evaluate the integral for each function and add or subtract as necessary
- The answer may be required in exact form
- If not, round to three significant figures (unless told otherwise)

6-2-5-cie-fig2-we-solution-part-1

6-2-5-cie-fig2-we-solution-part-2

It is possible, in subtraction questions, to combine the separate integrals into one
- This is possible when the limits for each function are often the same in subtraction questions

$V = π \int_{a}^{b} {y_{1}}^{2} d x - π \int_{a}^{b} {y_{2}}^{2} d x = π \int_{a}^{b} ({y_{1}}^{2} - {y_{2}}^{2}) d x$

This doesn’t really apply to addition questions as if the limits are the same, you would be adding some of the same volume twice
If in any doubt avoid this approach as accuracy is far more important

Adding and Subtracting Volumes (CIE A Level Maths: Pure 1)