AP Calculus BC 8.2 Connecting Position Velocity and Acceleration of Functions Using Integrals Questions

Review motion with integrals by separating displacement from total distance and using velocity or acceleration data correctly.

Covered syllabus points

CHA-4.C—Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion

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Johanna jogs along a straight path. For $0 \leq t \leq 40$ , Johanna's velocity is given by a differentiable function v. Selected values of v(t), where t is measured in minutes and v(t) is measured in meters per minute, are given in the table above.

Using correct units, explain the meaning of the definite integral $\int_{0}^{40}|v(t)| d t$ in the context of the problem. Approximate the value of $\int_{0}^{40}|v(t)| d t$ using a right Riemann sum with the four subintervals indicated in the table.

$\int_{0}^{40}|v(t)| d t$ is the total distance Johanna jogs, in meters, over the time interval $0 \leq t \leq 40$ minutes.

\begin{aligned} \int_{0}^{40}|v(t)| d t & \approx 12 \cdot|v(12)|+8 \cdot|v(20)|+4 \cdot|v(24)|+16 \cdot|v(40)| \\ & =12 \cdot 200+8 \cdot 240+4 \cdot 220+16 \cdot 150 \\ & =2400+1920+880+2400 \\ & =7600 \text { meters } \end{aligned}

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Covered syllabus points

Question 1