Question

Evaluate the integral: \[ \int_{-3}^{3} \left\lfloor x \right\rfloor \, dx \]

Hint:

For integrals involving piecewise functions like \(\left\lfloor x \right\rfloor\), symmetry can simplify the evaluation, especially if the function is odd.

Answer 1

The function \(\left\lfloor x \right\rfloor\) represents the greatest integer less than or equal to \(x\). Since the integral is over a symmetric interval, and the floor function is an odd function, the areas on either side of the origin cancel out. Thus, the integral evaluates to 0.