Volumes of revolution of parametrically defined curves

Volumes of revolution involve determining the volume created by a curve rotated about either the x-axis or y-axis of a graph by integrating the area swept out by the curve.
Parametrically defined curves involve equations where both x and y are expressed in terms of a separate parameter, usually written as t. For example, the equations x = t^2 and y = t^3, define a parametric curve.
The volume of revolution of a parametrically defined curve about the x-axis can be given by the formula ∫πy^2dx (integral of pi times y squared with respect to x) from x=a to x=b, where a and b are the x-coordinates of the limits of revolution.
Similarly, the volume of revolution of a parametrically defined curve about the y-axis can be given by the formula ∫πx^2dy (integral of pi times x squared with respect to y) from y=c to y=d, where c and d are the y-coordinates of the limits of revolution.
These two formulas assume right circular cones whose vertices lie on the axis of rotation, not the general case. The cross-sectional area of such a cone at a given point is proportional to the square of its distance from the vertex.
To transform these x and y-axis revolution formulas to parametric form, simply substitute x and y with their respective parametric expressions in terms of t. The differential dx or dy becomes dt, with the limits adjusted to the corresponding parameter values.
For instance, if x = t^2 and y = t^3 with t in the interval [p, q], the volume of the solid formed by rotating the curve about the x-axis is ∫π(t^3)^2dt from t=p to t=q. For the y-axis, it is ∫π(t^2)^2dt from t=p to t=q.
Be aware that certain problems may require the use of techniques such as the disc method or shell method. In these cases, the formula may need to be modified to account for the given shape of the solid.
It’s crucial to sketch the given parametric curve for visual understanding before rotating it about any axis. This can significantly assist with identifying the limits of integration.