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Definite Integrals

The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

Level 2

         

Let \(f(x)\) be a real-valued function continuous on \(\left[0,2\right]\) such that \(f(x)=f(2x)\) for all \(x\). If \[\int_0^1 f(x) dx = 100,\] then what is the value of \[\int_0^2 f(x)dx ?\]

Let \(a,b,c\) be non-zero real constants such that \[\displaystyle \int_{0}^3 (3ax^2+2bx+c) \, dx=\displaystyle \int_{1}^3 (3ax^2+2bx+c) \, dx.\] What is the value of \(a+b+c\)?

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

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 \(f(x)= \frac {1}{\ln(x)}\).

\[\displaystyle \int_{0}^{12} \Bigl( \{x\}^2+\lfloor x \rfloor ^2\Bigr) \ dx = \ ? \]

Details and assumptions:

  • Every \(x\in \mathbb{R}\) can be written as \(x=\lfloor x \rfloor + \{x\} \).
  • As usual, \(\lfloor x \rfloor\) denotes greatest integer less than or equal to \(x\).
  • \(\{x\} \) is the fractional part of \(x\).

If \( \int_0 ^ {3} f(x) \, dx = 2 \), what is the value of

\[ \int_0^{3} \left( 3 - f(x) \right) \, dx ? \]

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