Determine the length of r = θ 0 ≤ θ ≤ 1 . This is polar coordinates.

Question

Determine the length of r = θ 0 ≤ θ ≤ 1

Answer 1

1. Substitution:

- θ = tan(x)

- dθ = sec²(x) dx

2. Change of Limits:

- For θ = 0, x = 0

- For θ = 1, x = π/4

3. Integral Transformation:

- √(θ² + 1) = √(tan²(x) + 1) = √(sec²(x)) = |sec(x)| = sec(x)

4. Transformed Integral:

- L = ∫ sec³(x) dx over [0,π/4]

5. Solving the Integral:

- The integral of sec³(x) is:

- ½(sec(x)tan(x) + ln|sec(x) + tan(x)|) + C

6. Applying Limits:

- L = ½[sec(π/4)tan(π/4) + ln|sec(π/4) + tan(π/4)|] - ½[sec(0)tan(0) + ln|sec(0) + tan(0)|]

- L = ½(√2 + ln(1 + √2))

Therefore, the solution is:

L = ½(√2 + ln(1 + √2))

Answer 2

r=Theta =T

it's a spiral, graphically, starts at the origin, heads east, turning northeast

(0,0),to (1,1) in polar coordinates

for r=T, 0<r<1, 0<T<1

(0,0) to about (.54,.84) in rectangular coordinates

(x,y) =(r, T) in polar

(x,y) =(cos1, sin1) when r=1, where T is in radians, 1 rad = 180/pi degrees = about 57.3 degrees

cos1=about .54 = cos(57.3 degrees)

sin1 = about .84 = sin(57.3 degrees)

(x,y) = (rcosT, rsinT)

x=rcosT, y=rsinT

r^2=x^2 + y^2

r=T

sqr(x^2+y^2) = T = arctan(y/x)

y/x = tan[(x^2+y^2)^.5]

.84/.54=1.6

= tan(1 rad.)=tan(57.3 degrees) =1.6

y=xtan[(x^2+y^2)^.5] is r=T in rectangular coordinates

for (0,0)

0=0tan[(0^2+0^2)^.5=0(0)=0

integrate in rectangular coordinates and the solution should be the same as in polar coordinates