\(\displaystyle f(x)=\left\{\begin{array}{cl}x,&-2\leq x<1\\x^2,&1\leq x<2\\(x-4)^2,&2\leq x\leq3\end{array}\right.\)일 때 \(\displaystyle\int_{-2}^3 f(x)dx\)를 구하라.
풀이
\(\displaystyle\int_{-2}^3 f(x)dx=\int_{-2}^1 xdx+\int_1^2 x^2dx+\int_2^3 (x-4)^2dx\)
\(\displaystyle=\left[\frac{x^2}{2}\right]_{-2}^1+\left[\frac{x^3}{3}\right]_1^2+\left[\frac{(x-4)^3}{3}\right]_2^3\)
\(\displaystyle=\frac{1}{2}-2+\frac{8}{3}-\frac{1}{3}-\frac{1}{3}+\frac{8}{3}=-\frac{3}{2}+\frac{14}{3}\)
\(\displaystyle=\frac{28-9}{6}=\frac{19}{6}\)