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Week 6, Long Day: Spatial Regions and Volume Integrals

We have over the past weeks integrated along curves (1-dimensional, 1D, geometry), over plane regions and surfaces (2D geometry), and we will today end with integration through spatial regions, also called solids (3D geometry). Again we will look at how to parametrize such solids and how to determine their corresponding Jacobians. Perhaps you can by now sense the common method of integration with just a different number of variables depending on the number of dimensions of the geometry.

In the coming weeks we will use our newly acquired knowledge about integration with multiple variables on issues concerning vector fields and flux. But for today, let’s finalize the skill of integration.

Today’s Key Concepts

Triple integral. Parametrization of spatial region or solid. The Jacobian $\mathrm{Jac}$ for 3D geometry. Volume integral. Volume calculation $\mathrm{Vol}(\Omega)$. Graph surface. Solid of revolution. Solid sphere and segments hereof.

Preparation and Syllabus

Today’s eNote is eNote 25 Surface and Volume Integrals.

Maple Ressources

Today’s Maple demo is 29_VolumeIntegrals.

Activity Program

  • 10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: $\,$ Your teachers are present

Group Exercises

  1. Warming up: A Tripple Integral
  2. Parametrized Spatial Regions and Their Jacobians
  3. The Unit Spehere
  4. Regions Bounded by a Graph Surface. By Hand and Maple
  5. Solid of Revolution. Parametrization and Integration
  6. Solid Spheres
  7. More About Spheres