Suppose y=v(t) represents the velocity of an object on an interval [a,b].

Using the Riemann sum approximation, n∑i=1|v(ti)|Δt , to approximate the area under the curve y=|v(t)| estimates which of the following?

Question

Suppose y=v(t)  represents the velocity of an object on an interval [a,b].

Using the Riemann sum approximation, n∑i=1|v(ti)|Δt   , to approximate the area under the curve y=|v(t)| estimates which of the following?

BlackTom AI · Accepted Answer

Consider the setup: y = |v(t)| is the speed of the object, and you are forming a Riemann sum n∑_{i=1}^n |v(t_i)| Δt to approximate the area under the curve y = |v(t)| over [a, b].

First, recall that the integral of |v(t)| with respect to t over [a, b] represents the total distance traveled by the object, because you accumulate the magnitude of velocity (speed) over time, regardless of direction.

Now examine the given option: it states that this Riemann sum estimates the total distance traveled by the object over the interval [a, b]. This aligns with the interpretation of the integral of speed as distance, since each term |v(t_i)|Δt contributes a nonnegative amount corresponding to how far the object travels in that subinterval, independent of whether the velocity is positive or negative.

If the velocity were not inside an absolute value, the integral could represent net displacement, which could be less than the total distance due to cancellations when the object reverses direction. By using |v(t)|, those sign cancellations are removed, and the sum measures the full path length.

Therefore, the Riemann sum approximates the total distance traveled, because it aggregates the nonnegative speed over time across the interval.

MATH_1225_17243_202501 5.1 Areas and Distances

Suppose y=v(t) represents the velocity of an object on an interval [a,b]. Using the Riemann sum approximation, n∑i=1|v(ti)|Δt , to approximate the area under the curve y=|v(t)| estimates which of the following?

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MATH_1225_17243_202501 5.1 Areas and Distances

Suppose y=v(t) represents the velocity of an object on an interval [a,b]. Using the Riemann sum approximation, n∑i=1|v(ti)|Δt , to approximate the area under the curve y=|v(t)| estimates which of the following?

View Explanation

Log in for full answers

Similar Questions

Match the type of estimation technique shown in the pictures with the correct label. 1: ____ 2: ____ 3: ____ 4: ____

Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

Express the following integral as a limit of Riemann sums:

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose represents the velocity of an object on an interval . Using the Riemann sum approximation, , to approximate the area under the curve estimates which of the following?

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 0 1 𝑥 2 𝑑 𝑥 = lim 1 𝑛 ( 1 2 𝑛 2 + 2 2 𝑛 2 + . . . + ( 𝑛 2 𝑛 2 ) ) = lim ∑ 𝑛 𝑚 𝑥 × 𝑛 2 2 3

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Match the type of estimation technique shown in the pictures with the correct label. 1: 2: 3: 4: