Calculus

# Integral Approximation - Increasing Intervals

Robert is approximating the integral $\displaystyle{\int_0^{2}8x^3dx}$ using a right Riemann sum. If Robert takes the limit of the right Riemann sum with infinitely many intervals, what is the resulting approximation?

Stephanie is approximating the integral $\displaystyle{\int_0^1 (2-x^2) dx}$ on $3$ intervals using either a Riemann sum or the Trapezium rule. What would happen if she uses $7$ intervals instead of $3?$

Max is applying the Trapezium rule to approximate $\displaystyle{\int_0^{\frac{\pi}{2}} 2\sin x dx}$ using $9$ intervals. What would happen if he uses $5$ intervals instead of $9?$

In using a Riemann sum to approximate an integral, the larger the number of intervals, the more accurate the approximation. If we are approximating $\displaystyle{\int_0^1(x^2+2)dx}$ using a left Riemann sum, what is the minimum number of intervals needed in order to make the error less than $0.1?$

Rebecca is approximating the integral $\displaystyle{\int_0^{4}(6x^2+2)dx}$ using the Trapezium rule. If Rebecca takes the limit of the number of intervals to infinity, what is the approximation she obtains?

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