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Consider the following statements: Statement 1: \(\int_{0}^{10} (x-\lfloor x\rfloor)\,dx = 5\) Statement 2: For a function \(f\) satisfying \(f(x+a)=f(x)\), the integral over one interval of length \(a\) repeats in the same way, i.e. the value of \(\int_a^n f(x)\,dx\) remains unchanged under this periodicity. Choose the correct option.

1. Statement 1 is incorrect, while Statement 2 is correct.
2. Both Statement 1 and Statement 2 are correct, and Statement 2 correctly explains Statement 1.
3. Both Statement 1 and Statement 2 are correct, but Statement 2 does not explain Statement 1.
4. Statement 1 is correct, while Statement 2 is incorrect.

Correct answer: Both Statement 1 and Statement 2 are correct, but Statement 2 does not explain Statement 1.

Solution

Statement 1 is correct because the integral of the fractional part of x over the interval from 0 to 10 evaluates to 5, as it represents the area under the sawtooth wave formed by the function. Statement 2 is also correct as it describes the periodic nature of functions, indicating that the integral over any interval of length a will yield the same result due to the function's periodicity, but it does not directly relate to the evaluation of Statement 1.

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