Volume of Revolution for a Curve Given in Parametric Form Revision Content

Understanding Volume of Revolution in Parametric Form

When a curve is given in parametric form, its x and y coordinates are expressed in terms of an additional parameter, typically denoted as ‘t’.
Volume of revolution refers to the volume of the three-dimensional figure that is formed when a two-dimensional shape is rotated about an axis. This is a common concept in both calculus and geometry, and is particularly useful in applications involving physics and engineering.
In the case of parametric equations, the object of interest is a curve defined by two functions, x(t) and y(t), that describe how the x and y coordinates of a point on the curve vary in terms of the parameter ‘t’. This curve in the x-y plane can be revolved around a line (such as the x-axis or y-axis) to generate a 3-dimensional shape.

The volume of the solid created by revolving a curve described parametrically between two points a and b around the x-axis is given by the formula: V = pi * ∫ (from a to b) [y(t)]² dx/dt dt
The volume of the solid created by revolving a curve described parametrically between two points a and b around the y-axis is given by the formula: V = pi * ∫ (from a to b) [x(t)]² dy/dt dt
In the above formulae, ∫ represents the integral from a to b, dx/dt and dy/dt represent the derivatives of the functions x(t) and y(t) with respect to t, and [y(t)]² and [x(t)]² represent the square of the parametric function y(t) and x(t) respectively.

First, express your function in parametric form.
Identify the limits of your integral (values for ‘a’ and ‘b’) and which axis you are revolving around.
Substitute your expressions for x(t), y(t), dx/dt or dy/dt into the appropriate volume formula.
Evaluate the integral to find the volume.

The formulas for volume of revolution are derived from similar concepts about revolving areas under curves about the x or y-axis in Cartesian coordinates. It may be helpful to understand these techniques first for curves expressed in Cartesian form.
Remember that the square of a function is always positive, which is significant when interpreting the meaning of volume.
Converting a complex integral to a simpler one is often key to solving problems of volume of revolution. This can sometimes be achieved through substitution or by simplifying the integrand.
Make sure to understand the fundamental theorem of calculus and how to apply it to solve the integral in the volume formulas.
Understanding the geometric interpretation of these mathematical formulas can provide a more intuitive approach to solving problems. Visualisation often helps to conceptualise the process of revolution and the generation of the 3D shape.

Be careful with your setup - identifying incorrect limits or revolving around the incorrect axis can lead to inaccurate results.
When expressing your function in parametric form, always ensure that x(t), y(t), dx/dt and dy/dt are all expressed correctly and in terms of ‘t’.
Pay attention to the rules for differentiation and integration, as mistakes in these steps can cause errors in your final answer.