Let f(x)=|x^ 2 -4| on the interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let f(x)=∣x2−4∣f(x)=|x^{2}-4| on the interval [−3,5][-3,5]. (1) Identify all xx in [−3,5][-3,5] where the definition of ff changes form, and list the subintervals on which ff can be written without absolute value symbols. (2) Write ∫−35f(x) dx\displaystyle \int_{-3}^{5}f(x)\,dx as the sum of definite integrals over the subintervals found in part (a), replacing f(x)f(x) by the appropriate algebraic expression on each subinterval. (3) Evaluate the integral in part (b) and hence calculate the average value of ff on [−3,5][-3,5]. Express the final answer as an exact number.

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