Question

Suppose (v(t)) represents the velocity of an object on an interval. Using the Riemann sum approximation (sum f(x_i) ⋅ Δ x_i), to approximate the area under the curve, estimates which of the following?

A. The displacement of the object
B. The total distance traveled by the object
C. The acceleration of the object
D. The instantaneous velocity of the object

Answer 1

Using the Riemann sum approximation on a velocity vs. time graph estimates the object's displacement, not the total distance traveled, the acceleration, or the instantaneous velocity.

Step-by-step explanation:

When using the Riemann sum approximation (sum f(x_i) ⋅ Δ x_i) to approximate the area under the curve of a velocity vs. time graph, this calculation estimates the displacement of the object. Displacement is represented by the area under the velocity-time curve. The Riemann sum approximates this area, which is the integral of the velocity function with respect to time. Since displacement accounts for direction, it should not be confused with the total distance traveled, which is the absolute value of the integral.

If the velocity is positive over the whole interval, the total distance traveled and the displacement are equal. However, if the velocity changes sign on the interval, indicating a change in direction, the total distance traveled is the sum of the absolute values of the areas under the curve, whereas the displacement is the algebraic sum.