Numerical Approximation of Integrals

Numerical Approximation of Integrals: Level 4 Challenges


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

01x3dx.\large \int_{0}^{1}x^3dx .

Give your answer to 4 decimal places.

S1=188722+188822++201322+201422 S2=188822+188922++201422+201522I=18872015 ⁣x22dx\begin{aligned} 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{aligned}

What can you say about the relative values of S1S_{1}, S2S_{2}, and II?

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

Given that a cubic polynomial f(x)f(x) that satisfy the system of equations above, find the value of 20132017f(x)dx \displaystyle \int_{2013}^{2017} f(x) \, dx .

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

11f(x)dx=f(k)+f(k)\int_{-1}^{1}f(x)dx = f(-k) + f(k)

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

  10×2e ⁣1ln(x)dx  = ?\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: e2.718e \approx 2.718, and ln(2)0.693\ln(2) \approx 0.693, and use the following graph of f(x)=1ln(x)f(x)= \frac {1}{\ln(x)}.

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