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Numerical Approximation of Integrals

Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.

Effects of Changing 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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