Mathematics

The average value of a function on an interval is found by integrating it and dividing by the interval's length. For $\sin(kx)$ on $[0,\pi]$, you need the integral $\int_0^\pi \sin(kx)\,dx$ to equal zero—meaning the positive and negative areas cancel out perfectly. This happens when the sine wave completes enough full cycles (or half-cycles) within the interval so that symmetry forces cancellation. Find the smallest $k$ where this balance occurs.

The average value of a function on an interval is found by integrating it and dividing by the interval's length. For $\sin(kx)$ on $[0,\pi]$, you need the integral $\int_0^\pi \sin(kx)\,dx$ to equal zero—meaning the positive and negative areas cancel out perfectly. This happens when the sine wave completes enough full cycles (or half-cycles) within the interval so that symmetry forces cancellation. Find the smallest $k$ where this balance occurs.