**Text:**Your task is to estimate how far an object traveled during the time interval \(0 \leq t \leq 8\), but you only have the following data about the velocity of the object:\[\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline\text{time (sec)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\hline\text{velocity (feet/sec)} & -4 & -2 & 3 & 4 & 2 & 3 & 1 & 3 & 1 \\\hline\end{array}\]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.**Graph:**- The graph is a visual representation of velocity over time.- The black curve represents the velocity at different time intervals, based on the provided data points.- Blue rectangles are drawn beneath the curve to represent the Left Endpoint Approximation.- The x-axis (t) represents time in seconds, ranging from 0 to 8.- The y-axis (y) represents velocity in feet per second, ranging from -5 to 5.**Description:**You decide to use a left endpoint approximation to estimate the total displacement. So, you pick up a blue pen and draw rectangles whose height is determined by the velocity at the left endpoint of each time interval. Each rectangle represents an estimate of the distance traveled during that time period. **Title: Estimating Total Displacement Using Endpoint Approximations**You decide to use a left endpoint approximation 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 approximation 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 approximation, find approximately how far the object traveled. Your answers must include the correct units.**- Total displacement = [____]- Total distance traveled = [____]---Using the same data, you also decide to estimate how far the object traveled using a right endpoint approximation. So, you sketch the curve again with a black pen, and draw rectangles whose height is determined by the velocity measurement at the right endpoint of each one-second interval.**Graph Explanation:**The graph provides a visual representation of the velocity of an object over time, with the y-axis representing velocity and the t-axis representing time in seconds. The curve indicates how the velocity changes over a nine-second interval. Blue rectangles, drawn at each interval's left endpoint, approximate the displacement. There is also an indication that a similar approach could be done using right endpoint approximations, though the specifics are not shown in this image. The shape and height of the rectangles closely follow the curve, showing changes at each one-second interval.

For left Riemann sum we need to use area of rectangle formed at left end point of each sub interval and for right Riemann sum we need to use area of rectangle formed at right end point of each sub interval . If rectangle formed lie below the x-axis then for displacement area have to be taken negative where as for distance all area have to be taken positive . As displacement can have negative value where as distance travelled is always positive end points of interval x0=0 , x1=1 , x2 =2 , x3=3 , x4= 4 , x5=5 , x6=6 , x7=7 , x8=8 Width of each sub interval ∆x=1 Left Riemann sum total displacement =∆x fx0+ fx1+fx2+fx3+fx4+fx5+fx6+fx7 Here f(x) denotes velocity of that object at that specified point =1×f0+ f1+ f2+ f3+ f4+ f5+ f6+ f7 = -4-2+3+4+2+3+1+3=10 So displacement = 10 unit Now total distance travelled =∆x fx0+ fx1+fx2+fx3+fx4+fx5+fx6+fx7 Here f(x) denotes absolute value of velocity ) at that specified point =1×f0+ f1+ f2+ f3+ f4+ f5+ f6+ f7 = -4+-2+3+4+2+3+1+3= 4+2+3+4+2+3+1+3=22 So total distance travelled = 22 unit Right Riemann sum total displacement =∆x fx1+fx2+fx3+fx4+fx5+fx6+fx7+fx8 Here f(x) denotes velocity of that object at right end of each sub intervals t =1× f1+ f2+ f3+ f4+ f5+ f6+ f7 +f8= -2+3+4+2+3+1+3+1=15 So displacement = 15 unit Now total distance travelled =∆x fx1+fx2+fx3+fx4+fx5+fx6+fx7+fx8 Here f(x) denotes absolute value of velocity ) at that specified point =1× f1+ f2+ f3+ f4+ f5+ f6+ f7 +f8= -2+3+4+2+3+1+3+1= 2+3+4+2+3+1+3+1=19 So total distance travelled = 19 unit