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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.

# Level 4

Using the Left-hand Rectangular Approximation Method and 11 rectangles, approximate

$\large \int_{0}^{1}x^3dx .$

$$\begin{eqnarray} S_{1}&=&\sqrt[22]{1887}+\sqrt[22]{1888}+\cdots+\sqrt[22]{2013}+\sqrt[22]{2014} \\ \\ \ S_{2}&=&\sqrt[22]{1888}+\sqrt[22]{1889}+\cdots+\sqrt[22]{2014}+\sqrt[22]{2015} \\ \\ I&=&\int_{1887}^{2015}\!\sqrt[22]{x}\,\mathrm{d}x \end{eqnarray}$$

What can you say about the relative values of $$S_{1}$$, $$S_{2}$$, and $$I$$?

$\large \begin{cases} {f(2013) = 3} \\ {f(2015) = 1} \\ {f(2017)=5} \\ {f(2019)=2015} \\ \end{cases}$

Given that a cubic polynomial $$f(x)$$ that satisfy the system of equations above, find the value of $$\displaystyle \int_{2013}^{2017} f(x) \, dx$$.

For all cubic polynomials $$f(x)$$, what positive value of $$k$$ makes the following statement true?

$\int_{-1}^{1}f(x)dx = f(-k) + f(k)$

Bonus question: What is this method of approximation known as?

$\bigg \lfloor \;10 \times \displaystyle \int_2^e \! \frac {1}{\ln(x)} \, \mathrm{d}x\;\bigg \rfloor = \ ?$

Details and Assumptions:

• You may use the following approximations: $$e \approx 2.718$$, and $$\ln(2) \approx 0.693$$, and use the following graph of $$f(x)= \frac {1}{\ln(x)}$$.
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