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Graphs
Published in W. Bolton, Mathematics for Engineering, 2012
It is often necessary to estimate the area enclosed between a graph line and the axis of a graph. For example, in estimating the work done when a spring is stretched through a distance x by a force F, the area under the graph line between extensions 0 and x is the work done. With a straight-line graph, the area can be easily calculated as, often, the sum of the areas of simple shapes such as rectangles and triangles. Where the graph is non-linear, methods that can be used include: Counting squaresThis involves counting the squares under the graph and then multiplying the number of squares by the value corresponding to their area. See Section 2.4.1 for more details.The mid-ordinate ruleThe area under the curve is divided into a number of equal width vertical strips and the area is then the sum of the mid-ordinate values of the strips multiplied by their width. See Section 2.4.2 for more details.The trapezoidal ruleThe area under the curve is divided into a number of equal width vertical strips and the area is then the product of the width of the strips and half the sum of the first and last ordinates plus the sum of the remaining ordinates. See Section 2.4.3 for more details.
Mathematical Operations
Published in Theodore Louis, Behan Kelly, Introduction to Optimization for Environmental and Chemical Engineers, 2018
The reader should note that the trapezoid rule is often the quickest but least accurate way to perform a numerical integration by hand. However, if the step size is decreased, the answer should converge – subject to round-off error – to the analytical solution. The results of each numerical integration must be added together to obtain the final answer for smaller step sizes.
Interpolation, Differentiation, and Integration
Published in Julio Sanchez, Maria P. Canton, Software Solutions for Engineers and Scientists, 2018
Julio Sanchez, Maria P. Canton
Other methods of integration have been developed that provide more accurate approximations than the trapezoidal rule. The major advantages of the trapezoidal rule are its ease of implementation, and that it can be used with tabular data in which the x-axis values are not equidistant.
Parsing AUC Result-Figures in Machine Learning Specific Scholarly Documents for Semantically-enriched Summarization
Published in Applied Artificial Intelligence, 2022
Iqra Safder, Hafsa Batool, Raheem Sarwar, Farooq Zaman, Naif Radi Aljohani, Raheel Nawaz, Mohamed Gaber, Saeed-Ul Hassan
Among the notable figure extraction systems, ChartSense (Jung et al. 2017) is a semi-automatic system that extracts data from charts. It uses a deep learning approach to classify the type of charts. However, it requires user interaction to complete the extraction task effectively. Likewise, Scatteract (Cliche et al. 2017) is a fully automated system that deals with scattered plots with a linear scale. This system uses deep learning for identifying the components of the charts and maps the pixels to the chart coordinates with the help of OCR and robust regression. They also focused on text detection, recognition and data extraction from scatterplots. To calculate the area under a curve, the trapezoidal rule has long been used. The trapezoidal rule is simply integral of the function, where the function is divided into small intervals, each representing a trapezoid (Tallarida and Murray 1987). A lot of work has been done using the trapezoidal rule to calculate the area under the curves for different fields. For example author states about using the Trapezoidal rule to calculate the area under discrete and continuous curves. The area under the curve approach is used in medicine, for instance, to calculate the magnitude of pain experienced by the patients or to compute the area for plasma level-time curve. However, to our knowledge, no work has been done to find the area under the curve of parsed result figures.
Buoyancy-Induced Convection of Alumina-Water Nanofluids in Laterally Heated Vertical Slender Cavities
Published in Heat Transfer Engineering, 2018
Massimo Corcione, Stefano Grignaffini, Alessandro Quintino, Elisa Ricci, Andrea Vallati
Once steadyFigure 2, Figure 3, Figure 4, Figure 5state is reached, the heat fluxes at the heated and cooled sidewalls, qh and qc, are obtained using the following expressions: in which the temperature gradients are evaluated by a second-order temperature profile embracing the wall-node and the two adjacent fluid nodes. The heat transfer rates added to the nanofluid by the heated sidewall and withdrawn from the nanofluid by the cooled sidewall, Qh and Qc, are then calculated as in which the integrals are computed numerically by means of the trapezoidal rule.
Buoyancy-Induced Convection in Water From a Pair of Horizontal Heated Cylinders Enclosed in a Square Cooled Cavity
Published in Heat Transfer Engineering, 2021
Marta Cianfrini, Massimo Corcione, Luca Cretara, Massimo Frullini, Emanuele Habib, Pawel Oclon, Alessandro Quintino, Vincenzo Andrea Spena, Andrea Vallati
Hence, the corresponding average Nusselt number of the i-th cylinder at time τ, [Nu(τ)]i, is calculated with the expression in which the integral is computed numerically by means of the trapezoidal rule.