AP Calculus BC 8.2 Motion with Integrals Overview

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

[Maximum number: 3]

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

8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals question 1