Calculus

Numerical Approximation of Integrals

Integral Approximation - Accuracy of Approaches

         

Suppose T,L,T, L, and RR are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral 01(16x2)dx\displaystyle{\int_0^{1}(16-x^2)dx} using 1919 intervals. Which of the following is the correct relationship between the three approximations?

Suppose T,L,T, L, and RR are the approximation values using the Trapezium rule, left Riemann sum, and right Riemann sum, respectively, of the integral 017(2x3+3)dx\displaystyle{\int_0^{17}(2x^3+3)dx} using 1515 intervals. Which of the following is the correct relationship between the three approximation values?

Which method gives the closest approximate value to the integral 044x2dx,\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 033x3dx\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 03(4x+2)dx\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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