Exam Questions - Volume of Revolution about the y-axis

Volume of Revolution About the Y-axis

The technique of volumes of revolution handles the calculation of volumes for 3D shapes that are formed by revolving a 2D plane region around a line (called the axis of revolution).
The process of revolution, in mathematical context, signifies spinning a 2D region around an axis. For example, spinning a semi-circle around its diameter creates a sphere.

The general equation for the volume revolution around the y-axis is given by: V = π ∫(x²) dy, where V is the volume, and the integral is evaluated from a to b - the lower and upper limits on the y-axis, which are respectively given by the curves.
This equation represents the sum of infinitesimally small discs with thickness dy, radius x, and area equal to πx². Their volume given by the formula for a cylinder’s volume, πr²h - in this case, πx²dy.

First, express x in terms of y by rearranging the equation of the curve.
Next, identify the limits of integration, a and b. These limits are the y-values where the enclosed region begins and ends respectively.
Substitute the expression of x in terms of y and the limits of integration into the volume formula and evaluate the integral.
The resulting value is the volume of the figure obtained by rotating the given region about the y-axis.

Sometimes, the equation of the curve is given as x = f(y). In such cases, we don’t need to rearrange the equation. However, if the equation is given as y = f(x), we need to rewrite it in terms of x = g(y) to accommodate the formula for volume of revolution about the y-axis.
Sketching the region to be revolved around the axis can help visualise the problem better.

Calculating volumes of revolution is a crucial mathematical technique that finds applications in various fields like physics, engineering, and architecture where it’s often necessary to calculate the volume of irregular shapes.
Familiarity with this method can also facilitate an understanding of advanced calculus, structural integrity of revolving bodies, and further mathematical modelling of real-world problems.