Exam Questions - Volume of revolution: parametric form

Exam Questions - Volume of revolution: parametric form

Volume of Revolution: Parametric Form

Basics of Volume of Revolution

  • The volume of revolution is the volume of a 3-dimensional solid created by revolving a 2D shape around an axis.
  • This technique is often used in calculus to determine volumes of solids that have circular cross sections.

Parametric Form

  • In the parametric form, we describe each variable x and y in terms of a third variable, usually denoted as t, also known as the parameter.
  • Parametric form is used when a relationship between x and y can’t be easily expressed or is more conveniently expressed in terms of a third variable.

Formula for Volume of Revolution in Parametric Form

  • The formula for the volume of revolution around the x-axis in parametric form is: V = ∫[a,b] (y(t))² dx/dt dt.
  • The formula for the volume of revolution around the y-axis in parametric form is: V = ∫[c,d] (x(t))² dy/dt dt.

Calculation of Volumes

  • To calculate volumes, first write the x and y coordinates in parametric form.
  • Then substitute these into your chosen volume formula and integrate over the given limits.

Example Problems

  • Be comfortable with a range of problems. In some cases, you may need to sketch the curve defined by the parametric equations first.
  • The limits of integration can usually be found by considering the range of t in which the curve is defined.

Partitioning Regions

  • In some problems, it might be necessary to partition the region into several sections and calculate the volume of revolution for each part.
  • Always add these volumes together to get the total volume.

Special Techniques

  • Special techniques such as trigonometric substitution or integration by parts might be required for certain problems.
  • Remain diligent in your algebra and maintain accuracy throughout your calculations.

Common Mistakes

  • Always remember to square the y-coordinate in the formula for revolving around the x-axis (and vice versa).
  • Don’t forget to include the differential from parametric derivative in the integrand of your volume formula.