Answer:
13
Step-by-step explanation:
The velocity is given by:
[tex]\displaystyle v(t)= -2t + 8[/tex]
And the total distance given the velocity is given by:
[tex]\displaystyle \int_{a}^{b}|v(t)|\, dt[/tex]
So, the total distance traveled by the particle from t = 1 to t = 6 will be:
[tex]\displaystyle d=\int_{1}^{6}|-2t+8|\, dt[/tex]
The absolute value tells us that:
[tex]\displaystyle |-2t+8| = \left\{ \begin{array}{ll} -2t+8 & \quad x \leq 4 \\ -(-2t+8) & \quad x > 4\\ \end{array} \right[/tex]
So, split the integral into two parts:
[tex]\displaystyle d=\int_{1}^{4} (-2t+8)\, dt+\int_{4}^{6}(-(-2t+8))\, dt[/tex]
Integrate:
[tex]\displaystyle d=(-t^2+8t)\Big|_{1}^{4}+(t^2-8t)\Big|_{4}^{6}[/tex]
Evaluate:
[tex]\begin{aligned} d&=[(-4^2+8(4))-(-1^2+8)]+[(6^2-8(6))-(4^2-8(4)]\\ &= [16-7]+[-12-(-16)]\\ &=9+4 \\ &=13 \end{aligned}[/tex]
