Chapter 12.3, Problem 17E

a.

To determine

To show: $r\left(t\right)=\cos \left(t\right)i+\sin \left(t\right)j+\left(1-\cos \left(t\right)\right)k$ , $0\le t\le 2\pi$ is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Also, find the equation for the cylinder and a plane.

b.

To determine

Sketch the unit tangent vectors at $t=0,\frac{\pi }{2},\pi \text{ and }\frac{3\pi }{2}.$

(c)

To determine

To show: acceleration component is parallel to the plane.

d.

To determine

Find the length of the ellipse using Integral.

e.

To determine

Estimate the integral $L={\displaystyle {\int }_0^{2\pi }\left(\sqrt{{\cos }^2(t)+2{\sin }^2(t)}\right)dt}$ using Numerical Integration.