Calculus

Integral Approximation - 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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