Integrals play a crucial role in calculation, especially when determining the average value of a function over a certain interval. The process begins with setting up a definite integral for the function over the defined interval. In this exercise, we aim to find the average value of \( f(x) = 4 - x^2 \) over \([-2, 2]\).
The formula to calculate the average value is:
- \( \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \)
Substituting \( a = -2 \) and \( b = 2 \) into the formula provides \( \frac{1}{4} \int_{-2}^{2} (4 - x^2) \, dx \). Solving this integral involves finding the antiderivative of \( f(x) \) and evaluating it from \(-2\) to \(2\).
Integration gives the total "area" under the curve over this period, and the division by the interval's length adjusts it for calculating the average. By understanding and using integrals, one can not only determine this average but also explore the area under curves that has wide applications in real-life scenarios, like physics and engineering.