Homework 411: Use Simpson's Rule To Find The Definite Integr

Home Work 411 Use Simpsons Rule To Find Out The Definite Integralq

This homework involves applying numerical methods to evaluate definite integrals and derivatives of functions. The tasks include using Simpson’s rule and the trapezoidal rule to approximate the definite integral of a specified function over an interval, verifying these results through analytical formulas, employing MATLAB to compute the same integral with built-in functions, and calculating derivatives using difference methods. The aim is to compare numerical approximation methods, understand their accuracy, and leverage computational tools for complex calculations.

Paper For Above instruction

Numerical integration and differentiation are fundamental tools in applied mathematics, engineering, and physical sciences, especially when dealing with functions that lack closed-form integrals or derivatives. This paper explores the application of Simpson’s rule, trapezoidal rule, and difference methods for derivatives in evaluating the integral and derivative of the function \(f(x) = \frac{1}{3}x^3 + x^2\) over specified intervals, complemented by MATLAB’s computational capabilities.

Application of Simpson’s Rule for Numerical Integration

Simpson’s rule is a highly accurate method for approximating the definite integral of a function using parabolic segments fit over subintervals. To apply Simpson’s rule over the interval \([-1, 5]\), the interval is subdivided into an even number of subintervals, say \(n\). The width of each subinterval is then \(h = (b - a)/n\), where \(a = -1\) and \(b = 5\).

For simplicity, assuming \(n=4\), the points are at \(x=-1, 0, 1, 2, 5\). Function evaluations are computed at these points: \(f(-1), f(0), f(1), f(2), f(5)\). The Simpson’s rule formula then becomes:

\(I \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\)

Calculating the function values at these points and applying the formula yields an approximation of the integral. Increasing the number of subintervals enhances the accuracy of the approximation.

Application of Trapezoidal Rule

The trapezoidal rule approximates the area under the curve as trapezoids over each subinterval. Using the same subdivision, the integral is estimated by:

\(I \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\)

Calculations at the same points as above provide another numerical estimate, which is generally less precise than Simpson’s rule but easier to compute.

Analytical verification

Using the integral table, the exact value of the integral can be derived by integrating the function analytically:

\(\int f(x) dx = \frac{1}{12} x^4 + \frac{1}{3} x^3 + C\)

Evaluating this at the limits \(-1\) and \(5\), yields the exact integral value, which serves to verify the accuracy of the numerical methods.

Computational Implementation using MATLAB

MATLAB offers built-in functions such as integral() and quad() to evaluate definite integrals efficiently and accurately. By defining the function \(f(x)\), MATLAB's integral() function computes the integral over \([-1, 5]\), providing a reference for the accuracy of manual calculations. For example:

f = @(x) (1/3)*x.^3 + x.^2;


Q = integral(f, -1, 5);

This approach simplifies integral evaluation and provides high-precision results suitable for comparison.

Derivative Calculation Using Difference Methods

The derivative of the function \(f(x) = \frac{1}{3}x^3 + x^2\) at specific points involves both analytical and numerical techniques. The analytical derivative is:

\(f’(x) = x^2 + 2x\)

for \(x=3.5\), this evaluates straightforwardly. For numerical approximations, three difference methods are employed:

  • Forward difference: \(f’(x) \approx \frac{f(x+h) - f(x)}{h}\)
  • Backward difference: \(f’(x) \approx \frac{f(x) - f(x-h)}{h}\)
  • Central difference: \(f’(x) \approx \frac{f(x+h) - f(x-h)}{2h}\)

Using a small increment, say \(h=0.01\), the function is evaluated at points \(x=h, x, x-h, x+h\) over the interval [1, 6], and the derivatives computed accordingly. MATLAB automates this process, enabling curve evaluations over [1, 6], followed by derivatives estimation using the above methods.

Conclusion

Numerical methods like Simpson’s rule and the trapezoidal rule provide effective means for approximating integrals, especially when analytical solutions are complicated or unavailable. Comparing these methods with MATLAB’s reliable built-in functions offers valuable insights into their accuracy and limitations. Similarly, difference methods for derivatives facilitate understanding of function behavior at specified points, with central difference typically providing the most accurate approximation among the three. Employing computational tools streamlines complex calculations, enhances precision, and supports verification of analytical results, contributing significantly to applied mathematical analyses in various scientific and engineering contexts.

References

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