1. **State the problem:** We are given a velocity function $v(t)$ over the interval $0 \leq t \leq 5$ with known displacement and total distance traveled. We need to find the value of the definite integral $$\int_2^4 v(t) \, dt$$. 2. **Given information:** - Displacement over $[0,5]$ is 3, i.e., $$\int_0^5 v(t) \, dt = 3$$. - Total distance traveled over $[0,5]$ is 13, i.e., $$\int_0^5 |v(t)| \, dt = 13$$. 3. **Understanding displacement and distance:** - Displacement is the net change in position, the integral of velocity. - Total distance is the integral of the absolute value of velocity. 4. **Analyze the velocity graph:** - Velocity is positive on $[0,2)$ and $[4,5]$. - Velocity is negative on $(2,4)$. 5. **Calculate integrals over subintervals:** Let $$A = \int_0^2 v(t) \, dt,$$ $$B = \int_2^4 v(t) \, dt,$$ $$C = \int_4^5 v(t) \, dt.$$ Then, $$A + B + C = 3$$ (displacement), $$A - B + C = 13$$ (total distance, since $v$ is negative on $[2,4]$, absolute value flips sign). 6. **Solve the system:** Subtract the first from the second: $$ (A - B + C) - (A + B + C) = 13 - 3 \Rightarrow -2B = 10 \Rightarrow B = -5.$$ 7. **Answer:** $$\int_2^4 v(t) \, dt = -5.$$