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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.

Accuracy of Approaches

         

Suppose \(T, L,\) and \(R\) are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral \(\displaystyle{\int_0^{1}(16-x^2)dx}\) using \(19\) intervals. Which of the following is the correct relationship between the three approximations?

Suppose \(T, L,\) and \(R\) are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral \(\displaystyle{\int_0^{17}(2x^3+3)dx}\) using \(15\) intervals. Which of the following is the correct relationship between the three approximation values?

Which method gives the closest approximate value to the integral \(\displaystyle{\int_0^{4}4x^2 dx},\) a right Riemann sum, a left Riemann sum, or the Trapezium rule, where the same number of intervals is used for each approximation?

Suppose the integral \(\displaystyle{\int_0^{3}3x^{3}dx}\) is approximated using a right Riemann sum and a left Riemann sum. Which sum gives a more accurate approximation to the integral?

Sarah is approximating the integral \(\displaystyle{\int_0^{3}(4x+2)dx}\) using a Riemann sum. Would a right Riemann sum or left Riemann sum give a more accurate approximation?

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