Volumes of Revolution Revision Notes for Edexcel A-Level Further Mathematics

5.1.2 Modelling with Volumes of Revolution

Introduction to Modelling with Volumes of Revolution

Volumes of revolution arise when a region in the plane is rotated about an axis, forming a three-dimensional solid. In mathematical modelling, these solids can represent objects such as bottles, domes, or pipes. Using calculus, we calculate the volume of these solids by integrating the squared radius of the rotation multiplied by $\pi$

Volume Formulas for Modelling

Rotation About the $x$ Axis

V = \pi \int_a^b [y(x)]^2 \, dx

This formula is used when the region is bounded by a curve $y = f(x)$ , the $x-axis$ , and vertical lines $x = a$ and $x = b$ .

Rotation About the $y$ Axis

V = \pi \int_a^b [x(y)]^2 \, dy

This formula applies when the region is bounded by a curve $x = g(y)$ , the y-axis, and horizontal lines $y = a$ and $y = b$ .

Worked Examples

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Example 1: Modelling a Wine Glass

Problem:

The curve $y = x^2$ (from $x = 0$ to $x = 2$ ) represents the profile of a wine glass.

Calculate the volume when the curve is rotated about the x-axis.

Solution:

Step 1: Set up the integral: The curve $y = x^2$ rotates about the $x$ -axis.

Use the formula:

V = \pi \int_a^b [y(x)]^2 \, dx

Here, $y(x) = x^2$ , $a = 0$ , $b = 2$ :

V = \pi \int_0^2 (x^2)^2 \, dx = \pi \int_0^2 x^4 \, dx

Step 2: Integrate:

V = \pi \left[ \frac{x^5}{5} \right]_0^2 = \pi \left( \frac{2^5}{5} - \frac{0^5}{5} \right) = \pi \times \frac{32}{5}

Step 3: Simplify:

V = \frac{32\pi}{5}

The volume of the wine glass is $\frac{32\pi}{5} \, \text{units}^3$

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Example 2: Modelling a Vase

Problem:

The curve $x = \sqrt{y}$ , for $y \in [1, 4]$ , represents the profile of a vase.

Calculate the volume when the curve is rotated about the $y$$-axis$ .

Solution:

Step 1: Set up the integral:

Use the formula for rotation about the $y$ -axis:

V = \pi \int_a^b [x(y)]^2 \, dy

Here, $x(y) = \sqrt{y}$ , $a = 1$ , $b = 4$ :

V = \pi \int_1^4 (\sqrt{y})^2 \, dy = \pi \int_1^4 y \, dy

Step 2: Integrate:

V = \pi \left[ \frac{y^2}{2} \right]_1^4 = \pi \left( \frac{4^2}{2} - \frac{1^2}{2} \right)

Step 3: Simplify:

V = \pi \left( \frac{16}{2} - \frac{1}{2} \right) = \pi \times \frac{15}{2}

The volume of the vase is $\frac{15\pi}{2} \, \text{units}^3$

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Example 3: Composite Solids

Problem:

A cylindrical pipe is modelled as the difference between two solids. The outer radius of the cylinder is given by $r_{\text{outer}} = 3$ , and the inner radius by $r_{\text{inner}} = 2$ . The length of the cylinder is $h = 10$ .

Find the volume of the hollow pipe.

Solution:

Step 1: Volume of outer solid:

V_{\text{outer}} = \pi r_{\text{outer}}^2 h = \pi (3)^2 (10) = 90\pi

Step 2: Volume of inner solid:

V_{\text{inner}} = \pi r_{\text{inner}}^2 h = \pi (2)^2 (10) = 40\pi

Step 3: Volume of the pipe:

V_{\text{pipe}} = V_{\text{outer}} - V_{\text{inner}} = 90\pi - 40\pi = 50\pi

The volume of the pipe is $50\pi \, \text{units}^3$

Note Summary

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Common Mistakes:

Forgetting to square the radius in the formula: Always square the $y(x)$ or $x(y)$ term before integrating.
Using the wrong limits of integration: Ensure the limits match the axis of rotation (e.g., $x-limits$ for $x-axis$ rotation).
Ignoring axis of rotation: Double-check whether the rotation is about the $x-$ or $y-axis$ .
Not simplifying composite solids: When subtracting volumes, ensure all parts of the calculation use consistent units and dimensions.
Approximating results prematurely: Always keep answers in exact terms (e.g., with $\pi$ ) unless explicitly asked for a decimal approximation.

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Key Formulas:

Rotation about the $x-axis$ :

V = \pi \int_a^b [y(x)]^2 \, dx

Rotation about the $y-axis$ :

V = \pi \int_a^b [x(y)]^2 \, dy

Volume of a cylinder:

V = \pi r^2 h

Composite volumes (subtraction):

V_{\text{composite}} = V_{\text{outer}} - V_{\text{inner}}

Converting limits of integration: If given y-limits but rotating about the $x-axis$ , substitute the bounds correctly based on the curve.

Volumes of Revolution (Edexcel A-Level Further Mathematics): Revision Notes

5.1.2 Modelling with Volumes of Revolution

Introduction to Modelling with Volumes of Revolution

Volume Formulas for Modelling

Rotation About the $x$ Axis

Rotation About the $y$ Axis

Worked Examples

Example 1: Modelling a Wine Glass

Example 2: Modelling a Vase

Example 3: Composite Solids

Note Summary

Common Mistakes:

Key Formulas:

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