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(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

(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 parallelogram in the -plane with vertices ,

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

Which integral computes the volume of the solid bounded by

Find the osculating plane to the curve , parametrized by

Which of the following integrals computes the distance traveled by a particle moving with position vector

The picture below illustrates level sets of which function:

(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

Which of the following is the tangent plane to the graph of the function

Consider a particle, which starts at the point , and travels one full revolution, counter-clockwise, around the unit circle. Use Green’s theorem to find the work done by the vector field