Integration with Mathematica

For Equation (A 10.29), working through the integrals with Mathematica gives... 

Mikhailov M. D. and R. M. Cotta, 2001, Integral Transform Method with Mathematica, http //lttc.com.ufrj.br... 

Use Simpson s rule with at least 10 panels to evaluate the following definite integrals. Use Mathematica to check your results. 

If you give Mathematica a definite integral with an integrand that has no indefinite integral in Mathematica s tables, Mathematica will simply return your input statement. To carry out a numerical approximation to the integral and obtain a numerical value, use the NIntegrate statement, which has the form ... 

The rate equations will be stated in these terms for a number of reactions. In all these cases, the integrands are ratios of second-degree equations. The moderately complex integrations are accomplished with the aid of a table of integrals, or by MATHEMATICA, or numerically when the constants are known. 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

In 62 the authors present a detailed Mathematica program which is used to integrate a radial Schrodinger equation by a new two-step multiderivative numerical method with fourth- and sixth-order derivatives. In the program the authors use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. 

Newton and the Spherical Earth. One of the first quantitative effective theories that I can think of is that associated with Newton and the invention of the integral calculus. In particular, the key question that had to be faced was whether or not it was possible to pretend as though the entire mass of the earth is concentrated at its center, rendering it for gravitational purposes, nothing more than a point mass. Newton s argument was schematized pictorially as in fig. 12.1 and asserted as Proposition LXXVI, Theorem XXXVI of the famed Principia Mathematica (Motte 1934). 

All the theoretical work was performed by making use of mixed symbolic-numerical computation via the Mathematica 7.0 platform [22], and a hybrid numerical-analytical methodology with automatic error control, the Generalized Integral Transform Technique - GITT [23-26], in handling the governing partial differential equations. 

Mathematica also provides us with the differential equation solver DSolve, which can be employed for this problem. When we do this we do not have to work quite as much as we did using the integration methods. The solution looks as follows ... 

Using Mathematica we have two primary choices on how to proceed with the solution to this equation—we can rearrange it into its separable components and then integrate both sides of the equation or we can solve it directly we will do the latter. 

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