This study is a work on analyzing the traditional shapes of classrooms. A model is presented that considers the rectangular classroom divided into five sections as a function of two factors: viewing distance and horizontal viewing angle. The reading time and number of errors are affected by both. This model is validated experimentally, and the shape of a trapezoid is suggested for new classrooms.

Some new inequalities are obtained relating to the generalized trapezoid and midpoint rules for the Riemann–Stieltjes integral with a convex integrand and monotone nondecreasing integrator. Results are deduced for the special case of weighted Riemann integrals. Mathematics subject classification (2000): Primary 26D15, Secondary 26D10..

In this paper we first derive a weighted Montgomery identity on time scales and then establish weighted Ostrowski, trapezoid and Grüss type inequalities on time scales, respectively. These results not only provide a generalization of the known results, but also give some other interesting inequalities on time scales as special cases.

We define the (n, i, f)-tube orders, which include interval orders, trapezoid orders, triangle orders, weak orders, order dimension n, and interval-orderdimension n as special cases. We investigate some basic properties of (n, i, f)-tube orders, and begin classifying them by containment.

Some Ostrowski and trapezoid type inequalities for the Stieltjes integral in the case of Lischitzian integrators for both Hölder continuous and monotoonic integrals are obtained. The dual case is also analysed. Applications for the midpoint rule are pointed out as well.