China
Africa
Explore chapters and articles related to this topic
Solution of the Elastic Problem
Published in Abdel-Rahman Ragab, Salah Eldin Bayoumi, Engineering Solid Mechanics, 2018
Abdel-Rahman Ragab, Salah Eldin Bayoumi
Many engineering problems are concerned with solids of revolution, such as cylinders, disks, and spherical shells. Due to symmetry about the axis of revolution, taken as the z-axis, the displacements, strains, and stresses are independent of θ; hence, v=0 and γrθ=γzθ=τrθ=τzθ=0
Differentiation and Integration
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
Solids of revolution are the result of revolving a curve around an axis and tracing out the solid that is created. This is the abstract analogue of clay on a pottery wheel. As the wheel turns and the clay is shaped, a rotationally symmetric solid is spun. The solid of revolution mirrors this process.
Pre-service elementary mathematics teachers’ methods when solving integral-volume problems and the rationale behind their selections
Published in International Journal of Mathematical Education in Science and Technology, 2023
This study aims to determine which method students choose in integral volume problems and whether technology-assisted education is effective against the background of these selections. Integral volume problems are considered within the scope of Calculus course at Education Faculty in Turkey. The topic of volume integral calculations, considered to be an application of definite integration, includes finding the volume of a solid of revolution. In order to find the volume of a solid of revolution, which is a solid figure obtained by rotating a plane curve around a straight line (known as an axis) that lies in the same plane, methods such as the disk method, washer method, and shell method must be used (Thomas et al., 2009). The factors that determine which method is chosen are the relative difficulty or ease of calculating the turning radius and whether vertical or horizontal rotation will be required. The volume of a solid of revolution, obtained by rotating a plane curve around the x or y-axis, can be easily calculated by using both the washer and shell methods, although this may not always be the case. For example, in the washer method, when a plane curve is rotated around the y-axis, we should integrate with respect to y, although we may not be able to express y as a function. In this case, the shell method allows us to integrate with respect to x instead of y (Thomas et al., 2009).
Representational fluency in calculating volume: an investigation of students’ conceptions of the definite integral
Published in International Journal of Mathematical Education in Science and Technology, 2022
A solid of revolution is generated by revolving a plane region about a line in the same plane. Students are introduced to the slicing method before calculating the volume of a solid of revolution in a typical calculus classroom. Usually, after some applications involving the volumes of some familiar solids, they switch to applications where the slices form disks, washers, or cylindrical shells. Students should utilize the visual representations of 2D and 3D mathematical objects to identify the location, radius, thickness, or height of these slices. Simultaneously, they need to use the corresponding mathematical expressions and symbols to write a formula to calculate the volume. Therefore, students trying to solve a volume of revolution problem by integration should establish the relationships between visual and algebraic reasoning. From the instructional perspective, similarly to what previous researchers have stated (e.g. Grundmeier et al., 2006; Von Korff & Rebello, 2012), they should understand and explain which expressions they are writing and why in their formulas.