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Câu hỏi: Cho hàm số $f\left( x \right)$ có đồ thị trên đoạn $\left[ -3;3 \right]$ là đường gấp khúc $ABC\text{D}$ như hình vẽ. Tính $\int\limits_{-3}^{3}{f\left( x \right)\text{d}x}$
A. $-\dfrac{5}{2}$.
B. $\dfrac{35}{6}$.
C. $-\dfrac{35}{6}$.
D. $\dfrac{5}{2}$
A. $-\dfrac{5}{2}$.
B. $\dfrac{35}{6}$.
C. $-\dfrac{35}{6}$.
D. $\dfrac{5}{2}$
Dựa vào đồ thị, ta xác định được $AB: y=x+3$, $BC: y=1$, $CD: y=-\dfrac{3}{2}x+\dfrac{5}{2}$
Suy ra $f\left( x \right)=\left\{ \begin{aligned}
& x+3\text{khi} -3\le x\le -2 \\
& 1\text{khi} -2\le x\le 1 \\
& -\dfrac{3}{2}x+\dfrac{5}{2}\text{khi} 1\le x\le 3 \\
\end{aligned} \right.$
Vậy $\int\limits_{-3}^{3}{f\left( x \right)\text{d}x}=\int\limits_{-3}^{-2}{\left( x+3 \right)\text{d}x}+\int\limits_{-2}^{1}{\text{d}x}+\int\limits_{1}^{3}{\left( -\dfrac{3}{2}x+\dfrac{5}{2} \right)\text{d}x}$ $=\dfrac{5}{2}$.
Suy ra $f\left( x \right)=\left\{ \begin{aligned}
& x+3\text{khi} -3\le x\le -2 \\
& 1\text{khi} -2\le x\le 1 \\
& -\dfrac{3}{2}x+\dfrac{5}{2}\text{khi} 1\le x\le 3 \\
\end{aligned} \right.$
Vậy $\int\limits_{-3}^{3}{f\left( x \right)\text{d}x}=\int\limits_{-3}^{-2}{\left( x+3 \right)\text{d}x}+\int\limits_{-2}^{1}{\text{d}x}+\int\limits_{1}^{3}{\left( -\dfrac{3}{2}x+\dfrac{5}{2} \right)\text{d}x}$ $=\dfrac{5}{2}$.
Đáp án D.