Finding Average Value of sin(x) on [0,π]

Real Analysis · Axiom Academy

EXAMPLE Finding the Average Value of on Using the average-value formula to compute and locate where the function attains it. Find the average value of on the interval , and determine where on the interval the function attains that average value. The shaded rectangle has width and height , so its area equals the area under the arch of . The arch crosses the average-value line at the two points where , symmetric about . Nice work — you computed the average value of on and found where the function attains it. The average-value formula extends the idea of an average to a continuous function: . The result is , which sits between 0 and the maximum value 1 — exactly where you'd expect an average to land. Geometric meaning: is the height of a rectangle of width with the same area as the region under the arch. The Mean Value Theorem for integrals guarantees a c with ; here there are two, symmetric about . The average value connects the integral back to actual outputs of the function — a recurring idea from calculus into real analysis.

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