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.

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?

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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 \(3?\)

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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 \(9?\)

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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 \(0.1?\)

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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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