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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.
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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?
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Find the approximate value of \(\displaystyle{\int_0^{2} 7x^{3}dx}\) using the Trapezium rule, by dividing the interval into \(4\) pieces.
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What is the Trapezium rule approximation to the definite integral \(\displaystyle{\int_{-1}^1 (10-2x^2)dx}\) using four intervals?
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Which of the following represents the approximation of \(\displaystyle{\int_{0}^{2}x^{4}dx}\) using the Trapezium rule?
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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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