Numerical Approximation of Integrals

Integral Approximation - Increasing Intervals


Robert is approximating the integral 028x3dx\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 01(2x2)dx\displaystyle{\int_0^1 (2-x^2) dx} on 33 intervals using either a Riemann sum or the Trapezium rule. What would happen if she uses 77 intervals instead of 3?3?

Max is applying the Trapezium rule to approximate 0π22sinxdx\displaystyle{\int_0^{\frac{\pi}{2}} 2\sin x dx} using 99 intervals. What would happen if he uses 55 intervals instead of 9?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 01(x2+2)dx\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?0.1?

Rebecca is approximating the integral 04(6x2+2)dx\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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