Student handout: The Cone

Vector Calculus II 2021
Students set up and compute a scalar surface integral.
What students learn
  • Practice with “Use what you know.”
  • Practice interpreting integrals as “chop, multiply, add.”
An ice cream cone is to be dipped in chocolate. The cone can be described by the equation \(z^2=9\,(x^2+y^2)\), with \(0\le z\le9\) and \(x\), \(y\), and \(z\) in centimeters. The dipping process is such that the resulting (surface) density of chocolate on the cone is given by \(\sigma=1-{z\over9}\) in grams per square centimeter. Find the total amount of chocolate on the cone.
(There is no ice cream on the cone!)

Learning Outcomes