91Ó°ÊÓ

A wheel of radius \(r\) rolls without slipping over a horizontal plane at a uniform velocity \(v\). Suppose at time \(t=0\) the uppermost point \(A\) of the wheel has coordinates \((0,2 r)\) in a Cartesian coordinate system whose \(x\)-axis lies in the plane and is directed along the velocity vector. a) Write the law of motion \(t \mapsto(x(t), y(t))\) of the point \(A\). b) Find the velocity of \(A\) as a function of time. c) Describe graphically the trajectory of \(A\). (This curve is called a cycloid.) d) Find the length of one arch of the cycloid (the length of one period of this periodic curve). e) The cycloid has a number of interesting properties, one of which, discovered by Huygens \({ }^{10}\) is that the period of oscillation of a cycloidal pendulum (a ball rolling in a cycloidal well) is independent of the height to which it rises above the lowest point of the well. Try to prove this, using Example 9. (See also Problem 6 of the next section, which is devoted to improper integrals.)

a) Show that if \(f \in \mathcal{R}[a, b]\), then \(|f|^{p} \in \mathcal{R}[a, b]\) for \(p \geq 0\). b) Starting from Hölder's inequality for sums, obtain Hölder's inequality for integrals: \(^{6}\) $$ \left|\int_{a}^{b}(f \cdot g)(x) \mathrm{d} x\right| \leq\left(\int_{a}^{b}|f|^{p}(x) \mathrm{d} x\right)^{1 / p} \cdot\left(\int_{a}^{b}|g|^{q}(x) \mathrm{d} x\right)^{1 / q} $$ if \(f, g \in \mathcal{R}[a, b], p>1, q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\) c) Starting from Minkowski's inequality for sums, obtain Minkowski's inequality for integrals: $$ \left(\int_{a}^{b}|f+g|^{p}(x) \mathrm{d} x\right)^{1 / p} \leq\left(\int_{a}^{b}|f|^{p}(x) \mathrm{d} x\right)^{1 / p}+\left(\int_{a}^{b}|g|^{p}(x) \mathrm{d} x\right)^{1 / p} $$ if \(f, g \in \mathcal{R}[a, b]\) and \(p \geq 1\). Show that this inequality reverses direction if \(0<\) \(p<1\). d) Verify that if \(f\) is a continuous convex function on \(\mathbb{R}\) and \(\varphi\) an arbitrary continuous function on \(\mathbb{R}\), then Jensen's inequality $$ f\left(\frac{1}{c} \int_{0}^{\mathrm{c}} \varphi(t) \mathrm{d} t\right) \leq \frac{1}{c} \int_{0}^{\mathrm{c}} f(\varphi(t)) \mathrm{d} t $$ holds for \(c \neq 0\).