기함수 최솟값 함수 넓이

$\displaystyle f ( x ) = {\begin{cases} ( x \leq 0 ) & - x ^{2} - 2 x \\ ( x \geq 0 ) & x ^{2} - 2 x \end{cases}}$ $\displaystyle f ( x + 1 ) = {\begin{cases} ( x \leq - 1 ) & - x ^{2} - 4 x - 3 \\ ( x \geq - 1 ) & x ^{2} - 1 \end{cases}}$ $\displaystyle g ( x ) = {\begin{cases} ( x \leq - \frac{3}{2} ) & f ( x ) \\ ( - \frac{3}{2} \leq x \leq 0 ) & f ( x + 1 ) \\ ( 0 \leq x \leq 1 ) & f ( 1 ) \\ ( 1 \leq x ) & f ( x ) \end{cases}}$ 두 곡선 $y = f ( x )$와 $y = g ( x )$로 둘러싸인 넓이를 $S$라 하면 $\displaystyle S = \int _{- \frac{3}{2}} ^{0} {\left\{ f ( x ) - f ( x + 1 ) \right\} dx + \int _{0} ^{1} {\left\{ f ( x ) - f ( 1 ) \right\} dx}}$ $\displaystyle = \int _{- \frac{3}{2}} ^{- 1} {\left\{ ( - x ^{2} - 2 x ) - ( - x ^{2} - 4 x - 3 ) \right\} dx}$ $\displaystyle + \int _{- 1} ^{0} {\left\{ ( - x ^{2} - 2 x ) - ( x ^{2} - 1 ) \right\} dx}$ $\displaystyle + \int _{0} ^{1} {\left\{ ( x ^{2} - 2 x ) - ( - 1 ) \right\} dx}$ $\displaystyle = \frac{1}{4} + \frac{4}{3} + \frac{1}{3} = \frac{23}{12}$ $\therefore p + q = 35$