We need to find the integral of sine function from negative pi to pi. Let's visualize this problem by looking at the sine curve over this interval.
The key insight is that sine is an odd function, meaning sin of negative x equals negative sin of x. For any odd function, the integral over a symmetric interval around zero is always zero because the positive and negative areas cancel out perfectly.
Looking at the graph, we can see that the area below the x-axis from negative pi to zero is exactly equal in magnitude to the area above the x-axis from zero to pi. Since one area is negative and the other is positive, they cancel each other out completely.
We can also solve this using the Fundamental Theorem of Calculus. The antiderivative of sine x is negative cosine x. We evaluate this at pi and negative pi, then subtract. Since cosine of pi and cosine of negative pi both equal negative one, we get one minus one, which equals zero.
In conclusion, the integral of sine x from negative pi to pi equals zero. This result demonstrates the fundamental property that any odd function integrated over a symmetric interval around zero will always equal zero. Both the geometric interpretation and the analytical calculation confirm this answer.