"." (+1 ) dx (a) Find the Riemann sum for this integral using right endpoints and n = 4. 7 (b) Find the Riemann sum for this same integral, using left endpoints and n = 4 7\f10. 9.4 59.1 A graph of f is shown above. The numbers shown represent the geometric area of each region. Evaluate the following definite integrals. a) / f(x)dx = -82.6 o) ( fox)dx = -62.2 ( f(x)dx = -35.8 d) -5f(x) dx = 92.4\fThe table gives the values of a function obtained from an experiment. Use them to estimate ," f(x) dx using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. X 0 2 4 5 6 f(xx) 9.3 9.0 8.3 6.5 2.3 -7.6 -10.5 (a) R} = (b) L] = (c) M) = (d) If the function is known to be a decreasing function, can you say whether your estimates are less than or greater than the exact value of the integral? Less than the exact value # 1. R3 Greater than the exact value + 2. M} Less than the exact value 3. L3time (sec) 0 1 234567 8 velocity (feet/sec) -3 -4 -2 1 2 3 2 4 2 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. - --110 1.0 Left endpoint approximation 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 = -12fr Total distance traveled = 18fUsing the same data, you also decide to estimate how far the object traveled using a right endpoint Riemann sum. So, you sketch the curve again with a black pen, and height is determined by the velocity measurement at the right endpoint of each one-second interval. 110 - 1.0 Right endpoint approximation (B) Using the right endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units. Total displacement = -4ft Total distance traveled = 18/1