# Calc 1 Riemann Sums w/ velocity and distance

• Wm_Davies
In summary, The task is to estimate how far an object traveled in the time interval 0<= t >= 8 using only the given data about its velocity. Using the left endpoint Riemann sum, the displacement is found to be -11 feet. However, to find the total distance traveled, the area under the curve must be calculated, taking into account that area is always positive. Once this is done, the total distance traveled is determined.
Wm_Davies

## Homework Statement

This is somewhat a repost... except I have figured out some of it and I have cleaned up the question.

Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object.

$$\frac{time (sec)}{velocity (feet/sec)}\frac{0}{4}\frac{1}{1}\frac{2}{-2}\frac{3}{-3}\frac{4}{-4}\frac{5}{-3}\frac{6}{-1}\frac{7}{-3}\frac{8}{-1}$$

"See the attached graph."

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

Total displacement = "I have 11ft which is the right answer."

Total distance traveled = "I cannot figure this out"

## Homework Equations

Distance = time * velocity
Displacement = time * velocity

## The Attempt at a Solution

So I went ahead and got the Riemann sum of the left endpoint on the graph below.

$$\Delta$$X = 1

So I just added the y values.

The sum added up to -11 which was the answer for the displacement. I do not know why this is not the answer for the total distance but maybe I am missing something elementary.

#### Attachments

• Graph.jpg
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Last edited:
To find the distance traveled find the area of the shaded region. Area is always positive BTW :D

computerex said:
To find the distance traveled find the area of the shaded region. Area is always positive BTW :D

I tried to compute the area, but I am not getting it. Also area if area is always positive then why would the area of a curve under the x-axis be negative?

Wm_Davies said:
I tried to compute the area, but I am not getting it. Also area if area is always positive then why would the area of a curve under the x-axis be negative?

Distance can never be negative.

computerex said:
Distance can never be negative.

O.k. that actually makes tons of sense (I figured I was making some elementary mistake). So, I added up the areas as positive numbers and everything was correct. Thanks for the help.

## 1. What is a Riemann Sum in Calculus?

A Riemann Sum is a method used in calculus to approximate the area under a curve by dividing the area into smaller, simpler shapes and adding their individual areas together. It is used to find the exact value of an integral by using rectangles to approximate the curve.

## 2. How is velocity related to Riemann Sums in Calculus?

Velocity is the rate of change of displacement with respect to time. In Riemann Sums, velocity is represented by the slope of the secant line connecting two points on the curve. By calculating the Riemann Sum of the velocity function, we can find the total distance traveled by an object over a given time interval.

## 3. What is the difference between left, right, and midpoint Riemann Sums?

Left Riemann Sums use the left endpoint of each subinterval to calculate the height of the rectangles, while right Riemann Sums use the right endpoint. Midpoint Riemann Sums, on the other hand, use the midpoint of each subinterval. This results in slightly different approximations of the area under the curve.

## 4. How do you calculate the Riemann Sum with velocity and distance?

To calculate the Riemann Sum with velocity and distance, you first need to divide the time interval into smaller subintervals. Then, use the velocity function to calculate the height of each rectangle. Finally, add up the areas of all the rectangles to get an approximation of the total distance traveled by the object over the given time interval.

## 5. What are some real-life applications of Riemann Sums with velocity and distance?

Riemann Sums with velocity and distance have various real-life applications, such as calculating the total distance traveled by a car during a road trip, finding the total work done by a moving object, or determining the total amount of water flowing in a river over a period of time.

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