2

Part of 1995 Putnam

Problems(2)

Source:

For what pairs of positive real numbers

(a,b)

does the improper integral

(1)

converge? \begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}

Putnamcalculusintegrationcollege contests

Source:

An ellipse, whose semi-axes have length

a

b

, rolls without slipping on the curve

y=c\sin{\left(\frac{x}{a}\right)}

a,b,c

related, given that the ellipse completes one revolution when it traverses one period of the curve?

Putnamconicsellipsetrigonometryintegrationcalculusfunction