Resources for Multivariable Calculus

Fall 2020

Graphing Resources

Cylindrical and Spherical

This is a link to a Mathematica worksheet that lets you play around with cylindrical coordinates. Here's one for spherical coordinates.

Selected written up solutions

Multiple Integrals

• Find the volume of the region bounded by $x = y^2+2z^2$ and $x = 2-y^2$.
  Solution

• Set up a triple integral that finds the volume of the region between $x^2+y^2+z^2 = 4$ and $z = \sqrt{x^2+y^2}$ using rectangular coordinates, cylinderical coordinates, and spherical coordinates. Evaluate one of these integrals to find the volume.
  Solution

• Compute $\iint_\mathcal{D} (x+3y)\;dxdy$ using the map $x = u-2v, y = v.$
  Solution

  Line Integrals

• Compute $\int_C xe^{z^2}\; ds$ over hte piecewise linear path from $(0, 0, 1)$ to $(0, 2, 0)$ to $(1, 1, 1)$.
  Solution

• Evaluate $\oint_C \sin x dx + z\cos y dy + \sin y dz$ where $C$ is the ellipse $4x^2 + 9y^2 = 36$ oriented counterclockwise.
  Solution

Scalar Line Integral

Vector Line Integral