M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§4.1 Determining distance traveled from velocity
Study notes for Determining distance traveled from velocity in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- If velocity $v(t)\ge0$, the distance traveled on $[a,b]$ is the area under the velocity graph. For simple shapes (rectangles, triangles, trapezoids) compute geometrically.
- When $v$ changes sign, area above the axis counts positive and below counts negative for net change in position; total distance adds the absolute areas.
- Approximate irregular areas with rectangles: this is the seed of the Riemann sum.
- Position is an antiderivative of velocity: if $s'=v$ then $s(b) - s(a) = $ net signed area under $v$.
Key formulas
Net change in position
$$s(b)-s(a) = \int_a^b v(t)\,dt$$
Total distance
$$D = \int_a^b |v(t)|\,dt$$
Worked example
$v(t) = t - 2$ m/s on $[0,4]$. Find the net change in position and the total distance.
Show the solution
The graph is a line crossing zero at $t=2$: a triangle of area $2$ below the axis and a triangle of area $2$ above. Net change $= -2 + 2 = 0$ m; total distance $= 2 + 2 = 4$ m.
Common mistakes
- Reporting net change when total distance is asked.
Exam tipSketch the velocity graph first; most 4.1 problems are areas of triangles and rectangles.
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