Let f : R → R be defined by <span class="math-tex">\(\rm f(x)=\left\{\begin{matrix} \rm x+2 &\rm if\ x<0 \\ \rm |x-2| &\rm if \ x\geq0 \end{matrix}\right.\)</span>. Find <span class="math-tex">\(\rm \int_{-2}^{\ \ 3} f(x)\ dx\)</span>.
\(\int_{-2}^{0}(x + 2)\,dx = \left[\frac{x^2}{2} + 2x\right]_{-2}^{0} = 0 - (2 - 4) = 2\)
\(\int_{0}^{3}|x - 2|\,dx = \int_0^2 (2 - x)\,dx + \int_2^3 (x - 2)\,dx = 2 + 0.5 = 2.5\)
Total = 2 + 2.5 = 4.5
Answer: (C) 4.5
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