导函数是否一定黎曼可积？
本文来自一个学生的提问。 当时想了想，预感有这样的处处可导，但导函数又不黎曼可积的函数，可不知如何构造。若不要求导函数有界，则有很多例子，但若...
Math
功率与加热温度的关系在3D打印机热床温度上的应用
一. 背景介绍 设计3D打印机的热床时，热床的升温能力关乎着打印的成功与否。过低的温度会造成打印件翘曲，导致打印失败。因此，热床是否能升温到所需...
W3-S1 Areas of Surfaces of Revolution and the Theorem of Pappus
周次: 3 日期: March 9, 2022 节次: 1 A curve sweeps out a surface in the space around is called a surface of revolution. The “area” of this surface depends on the length of the rope and the distance of each of its segments from the axis of revolution. In this section, we calculate the area of surface of...
W2-S2 Moments and Center of Masses
周次: 2 日期: March 4, 2022 节次: 2 Outline In this lecture, we discuss moments ( 质量矩）and centers of mass. Two concepts are related. In this lecture, we only deal with one- and two-dimensional problems (Essentially all the problems been dealt with are in one-dimension). Basic Principle for Center of Mass and Moment Definition. (Center of Mass) For...
W2-S1 Lengths of Plane Curves
周次: 2 备注: Attendance check 日期: March 2, 2022 节次: 1 Outline We knew what is meant by the length of a straight line segment, but without calculus, we have no precise notion of the length of a general winding curve. The idea of approximating the length of a curve has no difference with the idea of approximating an area or approximating...
W1-S2 Volumes by Slicing (cont.) , Volumes by Cylindrical Shells
周次: 1 日期: February 25, 2022 节次: 2 Outline For solids of revolution, the volume is relatively easy to find out. In section 6.1 and 6.2, we developed three method for finding volumes. The first two are based on the area of cross-sections, therefore these cases are direct applications of the formula $V = \int_a^b A(x) dx.$ The third case uses cylindrical shells for...