Riemann sums with velocity and distance.

In summary, the problem is to estimate the total displacement of an object during the time interval 0-8 seconds using a left endpoint Riemann sum. The only data provided is the velocity of the object at one-second intervals. The student uses a black pen to plot the data points and connects them with a curve to get an idea of the velocity function. They then use a blue pen to draw rectangles to represent the left endpoint Riemann sum, assuming that the velocity is approximately constant on each one-second interval. This may not be an accurate assumption since the velocity and acceleration likely change continuously over time. The problem also includes parts B and C, but the student is unsure how to start and hopes to figure it out with some assistance
  • #1
Wm_Davies
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Homework Statement



I really need help starting this problem as I am not sure what to do.


Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object.



time (sec) 0 1 2 3 4 5 6 7 8
velocity (feet/sec) 4 1 -2 -3 -4 -3 -1 -3 -1


To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below.

"See the graph pictured below."

You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue pen and draw rectangles whose height is determined by the velocity measurement at the left endpoint of each one-second interval. By using the left endpoint Riemann sum as an approximation, you are assuming that the actual velocity is approximately constant on each one-second interval (or, equivalently, that the actual acceleration is approximately zero on each one-second interval), and that the velocity and acceleration have discontinuous jumps every second. This assumption is probably incorrect because it is likely that the velocity and acceleration change continuously over time. However, you decide to use this approximation anyway since it seems like a reasonable approximation to the actual velocity given the limited amount of data.

(A) Using the left endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units.

Total displacement = "I am not even sure what this is referring to"

Total distance traveled =

There is a part B & C ,but I think if I can get some help figuring this out then I can figure out the rest by myself (I hope).

Homework Equations



Distance = Time X Velocity

The Attempt at a Solution



I have no attempts because I am not sure how to start this problem.
 

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  • #2
Oh, I forgot to mention that this is for calculus 1.
 

Related to Riemann sums with velocity and distance.

1. What is a Riemann sum?

A Riemann sum is a method of approximating the area under a curve by dividing the region into smaller rectangles and finding the sum of their areas.

2. How is velocity related to Riemann sums?

Velocity can be represented as the slope of a curve, which is equivalent to the derivative of the function. Riemann sums are used to estimate the area under a curve, which can then be used to calculate the distance traveled by an object with that velocity.

3. What is the significance of using Riemann sums in relation to distance?

Riemann sums allow us to approximate the distance traveled by an object with changing velocity. By dividing the area under the curve into smaller rectangles, we can get a more accurate estimate of the distance traveled.

4. Can Riemann sums be used for any type of velocity function?

Yes, Riemann sums can be used for any continuous velocity function. However, for functions with non-constant velocity, the accuracy of the approximation may vary depending on the size of the rectangles used.

5. How do Riemann sums with velocity and distance relate to real-world applications?

Riemann sums with velocity and distance are commonly used in physics and engineering to model the motion of objects. They can be used to calculate the distance traveled by a moving object, the work done by a force, and the total energy of a system.

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