How can I find this displacement?

[GaussianSurface]

Homework Statement

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.
*First image
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.
Total displacement=
Total distance travelled=

Homework Equations

Δx = (b - a)/ n
displacement = Xf- Xo
distance = velocity ⋅ time

The Attempt at a Solution

None yet[/B]

 

[Ray Vickson]

Homework Statement

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.
*First image
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.
Total displacement=
Total distance travelled=

Homework Equations

Δx = (b - a)/ n
displacement = Xf- Xo
distance = velocity ⋅ time

The Attempt at a Solution

None yet[/B]

What is stopping you from just completing the calculation? You have all the $\Delta x$ and $y_i$ values, and you have a formula for the answer.

BTW: the formula should read as $S = \sum_i y_i (x_i - x_{i-1})$; that is, you should have $y_i$ and not $f(y_i)$ in the sum.

 

[Mark44]

you should have $y_i$ and not $f(y_i)$ in the sum.

You could have $y_i$ or $f(x_i)$ in the sum, but not, as Ray said, $f(y_i)$.