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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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