Weekly Assignment 10

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 Archimedian spiral!).

Find the length of the curve for an integral number of turns n about the origin.

Integrate[Sec[x]^3,x]
Integrate[(1/(1-u^2))^2,u]
Integrate[(1/2-1/4u)/(1-u)^2+(1/2+1/4u)/(1+u)^2,u]
Integrate[Sqrt[1+θ^2],θ,Reals]
Integrate[Sqrt[1+θ^2],{θ,0,2Pi n},Reals]

nturns=10
lengthCurve[n_]:=Integrate[Sqrt[1+θ^2],{θ,0,2Pi n}]
TableForm[
Table[{n,lengthCurve[n],N[lengthCurve[n]]},{n,1,nturns,1}]
]
lengthCurve[n]

ArchimedeanLength[n_]:=n π (Sqrt[1+4 (n^2) (π^2)])+(1/2) ArcSinh[2 n π]

ArchimedeanLength[1.0]
ArchimedeanLength[2.0]
AndimedeanLength[n_]:=n π (Sqrt[1+4 (n^2) (π^2)])+1/4 Log[(Sqrt[(2 n π)^2+1]+2 n π)/(Sqrt[(2 n π)^2+1]-2 n π)]
AndimedeanLength[n]
AndimedeanLength[1.0]
AndimedeanLength[2.0]

Find the physical area of the region that is swept out after an integral number of turns. (You might use a circle to approximate, to check your answer.)

areaCurve[n_]:=Integrate[1/2 θ^2,{θ,2π (n-1),2π n}]
Limit[areaCurve[n]/ArchimedeanLength[n],n->Infinity]
Simplify[areaCurve[n]]
nturns=10
PolarPlot[ θ,{θ,0,nturns*2π}]
Show[
ListPlot[Table[{n,N[areaCurve[n]]/N[π(n*2π)^2]},{n,1,nturns,1}],PlotRange->{{0,nturns},{0,1}}],
Plot[1,{n,1,nturns}]
]

TableForm[Table[{
n,
areaCurve[n],
N[lengthCurve[n]],
N[lengthCurve[n]]/N[Sum[(2π)^2(i-1/2),{i,1,n}]],
N[areaCurve[n]]/N[π(n*2π)^2],
N[areaCurve[n]/lengthCurve[n]]
}
,{n,1,nturns,1}
],
TableHeadings->{{},{"N","Area","Length","Length/Circumference","Area/Circle","Area/Length"}}
]

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⟩.