Volume of Revolution about the y-axis

Volume of Revolution about the Y-axis Revision Content

Understanding Volume of Revolution about the Y-axis

  • The volume of revolution about the y-axis refers to the volume of a three-dimensional object that is generated when a two-dimensional shape, or region, on a graph is revolved 360 degrees around the y-axis.
  • This is a common concept explored in calculus and it uses the principles of integration to determine these volumes.

Formulating the Integral for Volume of Revolution about the Y-axis

  • To find the volume of a shape generated by revolving the region bounded by the curve y = f(x) from x = a to x = b around the y-axis, the integral should be set up as follows: V = 2π ∫ from a to b of x*f(x) dx.
  • This is known as the shell method or cylindrical shells method. The method works by summing up the volumes of many infinitesimally thin cylinders (shells), which together comprise the total volume.

Solving the Integral

  • The definite integral is then evaluated to calculate the volume. This process typically involves using the fundamental theorem of calculus, applying integration techniques, and substitution where required.

Using Geometric Shapes

  • If the shape being revolved can be recognized as a standard geometric shape such as a circle, cone, or cylinder, volume of revolution can be found using basic geometry. For instance, a semicircle revolved around the y-axis creates a sphere, and the volume can be calculated using the formula for the volume of a sphere.

Illustrating the Graph

  • It can be useful to sketch the region being revolved and the resulting solid to better visualize the problem. Label the bounds of integration clearly on the sketch.

Importance of The Bounded Region

  • The region that is being revolved around the axis is crucial. Be clear whether the region is above or below the curve, and within the bounds of integration.

Common Pitfalls

  • A common mistake while setting up the integral is to multiply by ‘x’ instead of ‘f(x)’ when applying the shell method. It’s also important to position the ‘x’ term outside the integral and ‘f(x)’ inside in this context.
  • It’s crucial to ensure that the function is properly set up for integration with respect to x.

Notations and Conventions

  • Even though the revolution is being done around the y-axis, the integration is performed with respect to x. This can seem counterintuitive at first, but is the standard notation.
  • Ensure the boundaries for the definite integral are aligned with the x-values of the region being revolved, even though the revolution is about the y-axis.

Useful Techniques

  • Prior knowledge of integration techniques, like substitution, integration by parts, and recognition of common integral forms, is assumed and will be necessary to solve the problems.
  • Understanding how to sketch and interpret graphs of functions can be very helpful in visualising the volume of revolution problems.