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Numerical Approximation of Integrals

Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.

Trapezium Rule

         

The following is Devin's approximation of an integral using the Trapezium rule: \[\frac{6}{5}\cdot\left(\left(\frac{3}{5}\right)^{5}+2\cdot\left(\frac{6}{5}\right)^{5}+2\cdot\left(\frac{9}{5}\right)^{5}+2\cdot\left(\frac{12}{5}\right)^{5}+\left(\frac{15}{5}\right)^{5}\right).\] Which of the following integrals is Devin approximating?

Find the approximate value of \(\displaystyle{\int_0^{2} 7x^{3}dx}\) using the Trapezium rule, by dividing the interval into \(4\) pieces.

What is the Trapezium rule approximation to the definite integral \(\displaystyle{\int_{-1}^1 (10-2x^2)dx}\) using four intervals?

Which of the following represents the approximation of \(\displaystyle{\int_{0}^{2}x^{4}dx}\) using the Trapezium rule?

Find the approximate value of \(\displaystyle{\int_0^1 (12x^2+2)dx}\) using the Trapezium rule, by dividing the interval into \(5\) parts.

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