Author: Anonymous

(12 points) Partial credit problem. You must upload a scan of your solution at the end of the exam. Please show all of your workings for full points. Let be the loop given by the intersection of the surface and the cylinder , with counter-clockwise orientation. Let

Let be the solid upper-half of the unit sphere in

Use the Divergence theorem to calculate the flux integral

(12 points) Partial credit problem. You must upload a scan of your solution at the end of the exam. Please show all of your workings for full points. A particle has the position function and is moving under an acceleration of

(12 points) Partial credit problem. You must upload a scan of your solution at the end of the exam. Please show all of your workings for full points. Let S be the surface given by the portion of the graph of  that lies above the 1/8th of the unit disk in the -plane which is illustrated below. Compute

Which integral computes the volume of the solid bounded by

Find the osculating plane to the curve , parametrized by

The picture below illustrates level sets of which function:

Which integral gives the surface area of the surface parametrized by

Compute the work done by the vector field in moving a particle along a curve which begins at and ends at.