Selected Homework 11:  Polar Coordinates (Answers)

Exercise 4.

[Maple Plot]

`a. Create a polar plot of  r = `*f(theta) = 1+sin(theta), ` `*theta = 0 .. 2*Pi

f(theta) = 1+sin(theta)

x(theta) = (1+sin(theta))*cos(theta)

y(theta) = (1+sin(theta))*sin(theta)

`b. Find critical points`, dy/dx = 0

`f '`(theta) = cos(theta)

`x '`(theta) = cos(theta)^2-sin(theta)-sin(theta)^2

`y '`(theta) = 2*cos(theta)*sin(theta)+cos(theta)

dy/dx = `y '`(theta)/`x '`(theta)

`y '`(theta) = 0

theta = 1/2*Pi, -1/6*Pi, -5/6*Pi

[x, y] = [0, 2], [1/4*3^(1/2), -1/4], [-1/4*3^(1/2), -1/4]

`c. Find (polar) arc length.`

1/2*`Arc Length` = Int((f(theta)^2+`f '`(theta)^2)^(1/2),theta = -1/2*Pi .. 1/2*Pi)

`Now,`*(f(theta)^2+`f '`(theta)^2)^(1/2) = ((1+sin(theta))^2+cos(theta)^2)^(1/2)

`` = (2+2*sin(theta))^(1/2)

`` = 2^(1/2)*(1+sin(theta))^(1/2)*(1-sin(theta))^(1/2)/(1-sin(theta)*``)^(1/2)

1/2*`Arc Length` = Int(2^(1/2)*(1+sin(theta))^(1/2)*(1-sin(theta))^(1/2)/(1-sin(theta)*``)^(1/2),theta = -1/2*Pi .. 1/2*Pi)

1/2*`Arc Length` = 2^(1/2)*Int(cos(theta)/(1-sin(theta))^(1/2),theta = -1/2*Pi .. 1/2*Pi)

1/2*`Arc Length` = 4

`Arc Length` = 8

`d. Find (polar) area.`

Area = Int(1/2*f(theta)^2,theta = 0 .. 2*Pi)

Area = Int(1/2*(1+sin(theta))^2,theta = 0 .. 2*Pi)

Area = 3/2*Pi

Exercise 5.

[Maple Plot]

`a. Create a polar plot of  r = `*f(theta) = 1+2*sin(theta), ` `*theta = 0 .. 2*Pi

f(theta) = 1+2*sin(theta)

x(theta) = (1+2*sin(theta))*cos(theta)

y(theta) = (1+2*sin(theta))*sin(theta)

`b. Find critical points`, dy/dx = 0

`f '`(theta) = 2*cos(theta)

`x '`(theta) = 2*cos(theta)^2-sin(theta)-2*sin(theta)^2

`y '`(theta) = 4*cos(theta)*sin(theta)+cos(theta)

dy/dx = `y '`(theta)/`x '`(theta)

`y '`(theta) = 0

cos(theta)*(4*sin(theta)+1) = 0

theta = 1/2*Pi, 3/2*Pi, arctan(1/15*15^(1/2))-Pi, -arctan(1/15*15^(1/2))

[x, y] = [0, 3], [0, 1], [-1/8*15^(1/2), -1/8], [1/8*15^(1/2), -1/8]

`c. Find (polar) arc length.`

1/2*`Arc Length` = Int((f(theta)^2+`f '`(theta)^2)^(1/2),theta = -1/2*Pi .. 1/2*Pi)

`Now,`*(f(theta)^2+`f '`(theta)^2)^(1/2) = ((1+2*sin(theta))^2+4*cos(theta)^2)^(1/2)

`` = (5+4*sin(theta))^(1/2)

1/2*`Arc Length` = Int((5+4*sin(theta))^(1/2),theta = -1/2*Pi .. 1/2*Pi), ` (Elliptic integral - hard)`

1/2*`Arc Length` = 6.682446612

`Arc Length` = 13.36489322

`d. Find (polar) area of inner loop.`

f(theta) = 0

1+2*sin(theta) = 0

theta = 7/6*Pi, 11/6*Pi

Area = Int(1/2*f(theta)^2,theta = 7/6*Pi .. 11/6*Pi)

Area = Int(1/2*(1+2*sin(theta))^2,theta = 7/6*Pi .. 11/6*Pi)

Area = -3/2*3^(1/2)+Pi

Exercise 6.

[Maple Plot]

`a. Create a polar plot of  r = `*f(theta) = sin(2*theta), ` `*theta = 0 .. 2*Pi

f(theta) = sin(2*theta)

x(theta) = sin(2*theta)*cos(theta)

y(theta) = sin(2*theta)*sin(theta)

`b. Find critical points`, dy/dx = 0

`f '`(theta) = 2*cos(2*theta)

`x '`(theta) = 4*cos(theta)^3-2*cos(theta)-2*sin(theta)^2*cos(theta)

`y '`(theta) = 6*sin(theta)*cos(theta)^2-2*sin(theta)

dy/dx = `y '`(theta)/`x '`(theta)

`y '`(theta) = 0

2*sin(theta)*(3*cos(theta)^2-1) = 0

theta = 0, arccos(1/3*3^(1/2)), Pi-arccos(1/3*3^(1/2)), Pi+arccos(1/3*3^(1/2)), 2*Pi-arccos(1/3*3^(1/2))

[x, y] = [0, 0], [.54436, .76979], [.54436, -.76979], [-.54436, -.76979], [-.54436, .76979]

`c. Find (polar) arc length.`

1/4*`Arc Length` = Int((f(theta)^2+`f '`(theta)^2)^(1/2),theta = 0 .. 1/2*Pi)

`Now,`*(f(theta)^2+`f '`(theta)^2)^(1/2) = (sin(2*theta)^2+4*cos(2*theta)^2)^(1/2)

`` = (3*cos(2*theta)^2+1)^(1/2)

1/4*`Arc Length` = Int((3*cos(2*theta)^2+1)^(1/2),theta = 0 .. 1/2*Pi)

1/4*`Arc Length` = 2.422112054

`Arc Length` = 9.688448216

`d. Find (polar) area.`

Area = 4*Int(1/2*f(theta)^2,theta = 0 .. 1/2*Pi)

Area = 4*Int(1/2*sin(2*theta)^2,theta = 0 .. 1/2*Pi)

Area = 1/2*Pi

Exercise 7.

[Maple Plot]

`a. Create a polar plot of  r = `*f(theta) = cos(2*theta), ` `*theta = 0 .. 2*Pi

f(theta) = cos(2*theta)

x(theta) = cos(2*theta)*cos(theta)

y(theta) = cos(2*theta)*sin(theta)

`b. Find critical points`, dy/dx = 0

`f '`(theta) = -2*sin(2*theta)

`x '`(theta) = -6*sin(theta)*cos(theta)^2+sin(theta)

`y '`(theta) = -4*sin(theta)^2*cos(theta)+2*cos(theta)^3-cos(theta)

dy/dx = `y '`(theta)/`x '`(theta)

`y '`(theta) = 0

-cos(theta)*(4*sin(theta)^2-2*cos(theta)^2+1) = 0

cos(theta)*(3*cos(2*theta)-2) = 0

theta = 1/2*Pi, 3/2*Pi, .4205343352, Pi-.4205343352, Pi+.4205343352, 2*Pi-.4205343352

[x, y] = [0, -1], [0., 1.], [.60858, .27216], [-.60858, .27216], [-.60858, -.27216], [.60858, -.27216]

`c. Find (polar) arc length.`

1/4*`Arc Length` = Int((f(theta)^2+`f '`(theta)^2)^(1/2),theta = -1/4*Pi .. 1/4*Pi)

`Now,`*(f(theta)^2+`f '`(theta)^2)^(1/2) = (cos(2*theta)^2+4*sin(2*theta)^2)^(1/2)

`` = (-3*cos(2*theta)^2+4)^(1/2)

1/4*`Arc Length` = Int((-3*cos(2*theta)^2+4)^(1/2),theta = -1/4*Pi .. 1/4*Pi)

1/4*`Arc Length` = 2.422112054

`Arc Length` = 9.688448216

`d. Find (polar) area.`

Area = 4*Int(1/2*f(theta)^2,theta = -1/4*Pi .. 1/4*Pi)

Area = 4*Int(1/2*cos(2*theta)^2,theta = -1/4*Pi .. 1/4*Pi)

Area = 1/2*Pi

Exercise 8.

[Maple Plot]

`a. Create a polar plot of  r = `*f(theta) = sin(4*theta), ` `*theta = 0 .. 2*Pi

f(theta) = sin(4*theta)

x(theta) = sin(4*theta)*cos(theta)

y(theta) = sin(4*theta)*sin(theta)

`b. Find critical points`, dy/dx = 0

`f '`(theta) = 4*cos(4*theta)

`x '`(theta) = 32*cos(theta)^5-32*cos(theta)^3+4*cos(theta)-8*sin(theta)^2*cos(theta)^3+4*sin(theta)^2*cos(theta)

`y '`(theta) = 40*sin(theta)*cos(theta)^4-36*sin(theta)*cos(theta)^2+4*sin(theta)

dy/dx = `y '`(theta)/`x '`(theta)

`y '`(theta) = 0

4*sin(theta)*(10*cos(theta)^4-9*cos(theta)^2+1) = 0

theta = 0, .4999939881, arccos(1/3*3^(1/2)), Pi+arccos(1/3*3^(1/2)), 2*Pi-arccos(1/3*3^(1/2))

`` = 1.202165650, Pi-1.202165650, Pi+1.202165650, 2*Pi-1.202165650

[x, y] = [0, 0], [.79799, .43594], [-.36286, -.51316], [.36286, .51316], [.36286, -.51316]

`` = [-.35864, -.92850], [-.35864, .92850], [.35864, .92850], [.35864, -.92850]

`c. Find (polar) arc length.`

1/8*`Arc Length` = Int((f(theta)^2+`f '`(theta)^2)^(1/2),theta = 0 .. 1/4*Pi)

`Now,`*(f(theta)^2+`f '`(theta)^2)^(1/2) = (sin(4*theta)^2+16*cos(4*theta)^2)^(1/2)

`` = (15*cos(4*theta)^2+1)^(1/2)

1/8*`Arc Length` = Int((15*cos(4*theta)^2+1)^(1/2),theta = 0 .. 1/4*Pi)

1/8*`Arc Length` = 2.144605444

`Arc Length` = 17.15684355

`d. Find (polar) area.`

Area = 8*Int(1/2*f(theta)^2,theta = 0 .. 1/4*Pi)

Area = 8*Int(1/2*sin(4*theta)^2,theta = 0 .. 1/4*Pi)

Area = 1/2*Pi