Numerical Approximation of Integrals
Using the Left-hand Rectangular Approximation Method and 11 rectangles, approximate
\[\large \int_{0}^{1}x^3dx .\]
Give your answer to 4 decimal places.
by
Andrew Ellinor
\(\begin{eqnarray} S_{1}&=&\sqrt[22]{1887}+\sqrt[22]{1888}+\cdots+\sqrt[22]{2013}+\sqrt[22]{2014} \\ \\ \ S_{2}&=&\sqrt[22]{1888}+\sqrt[22]{1889}+\cdots+\sqrt[22]{2014}+\sqrt[22]{2015} \\ \\ I&=&\int_{1887}^{2015}\!\sqrt[22]{x}\,\mathrm{d}x \end{eqnarray} \)
What can you say about the relative values of \(S_{1}\), \(S_{2}\), and \(I\)?
by
Utsav Banerjee
\[ \large \begin{cases} {f(2013) = 3} \\ {f(2015) = 1} \\ {f(2017)=5} \\ {f(2019)=2015} \\ \end{cases} \]
Given that a cubic polynomial \(f(x) \) that satisfy the system of equations above, find the value of \( \displaystyle \int_{2013}^{2017} f(x) \, dx \).
by
Otto Bretscher
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?
by
Andrew Ellinor
\[\bigg \lfloor \;10 \times \displaystyle \int_2^e \! \frac {1}{\ln(x)} \, \mathrm{d}x\;\bigg \rfloor = \ ? \]
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 \(f(x)= \frac {1}{\ln(x)}\).
by
Utsav Banerjee