# The length of the curve r = cos(θ) - sin(θ), 0<θ<∏/4

1. May 19, 2014

### mill

1. The problem statement, all variables and given/known data

Find the length of the curve r = cos(θ) - sin(θ), 0<θ<∏/4

2. Relevant equations

L=∫ds
ds=sqrt(r^2 + (dr/dθ)^2)

3. The attempt at a solution

r^2 = (cosθ - sinθ)^2 = cos^2(θ) -2cosθsinθ + sin^2(θ) =1-2cosθsinθ

dr/dθ = -sinθ -cosθ

(dr/dθ)^2 = 1+2cosθsinθ

L=∫sqrt(2)dθ=sqrt(2)[θ]=sqrt(2) [pi/4]

But the answer is pi/(2*sqrt(2)). Where did I go wrong?

2. May 19, 2014

### jbunniii

You didn't. The two answers are the same, just written differently:
$$\frac{\sqrt{2} \pi}{4} = \frac{\sqrt{2} \pi}{4} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\pi}{4 \sqrt{2}} = \frac{\pi}{2 \sqrt{2}}$$

3. May 19, 2014

Thanks!