Weekly Assignment 11

MAT 229, Spring 2021

Instructions:  Show your work!

1. Polar curves

A. Consider the two polar curves r=1 and r=2+2cos(θ).

Find polar coordinates for all points of intersection.

Find the area of the region that is inside r=2+2cos(θ) and outside r=1.

B. Consider the polar curve r=θ (An Archimedean spiral).

Find the length of the curve  swept out after n complete rotations from angle 0; your answer should be a formula involving n.

Find the area of the region that is swept out over the last complete rotation when using n complete rotations from angle 0; your answer should be a formula involving n. (You might use a circle as an approximation to check your answer.)

2. Shapes in space

Let P be the point with Cartesian coordinates (2,1,4) and Q be the point (4,3,10).

What is the distance between them?

What are the coordinates for the midpoint of the line segment (OverBar[P Q])?

Find an equation for the sphere that has a diameter with one endpoint at P and the other at Q.

3. Unit vectors

Find the two unit vectors that are parallel to vector ⟨2,6⟩.