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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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