Your velocity is given by v(t)=2t2+5 in m/sec, with t in seconds. Estimate the distance, s, traveled between t=0 and t=8. Use the average of the left and right sums with 4 subdivisions.
The first step is to make a table. there are 5 even subdivisions between t=0 and t=8.
0 2 4 6 8
5 13 37 77 133
Left Hand: (5*2)+(13*2)+(37*2)+(77*2) = 264
Right Hand: (13*2)+(37*2)+(77*2)+(133*2) = 520
Add the left and right-hand sums together then divide by 2 to find the average.
Answer: s=392
To estimate the distance traveled using the average of the left and right sums with 4 subdivisions, we will divide the time interval [0, 8] into four equal subintervals.
Step 1: Determine the width of each subdivision.
The width of each subdivision is given by the total time interval divided by the number of subdivisions. In this case, the width of each subdivision is (8-0)/4 = 2 seconds.
Step 2: Calculate the left and right sums for each subdivision.
The left sum for each subdivision estimates the distance traveled during that time interval using the velocity at the left endpoint. The right sum estimates the distance using the velocity at the right endpoint.
For the left sum, we use the velocity at t=i, where i represents the left endpoint of the subdivision.
For the right sum, we use the velocity at t=i+1, where i represents the left endpoint of the subdivision.
Let's calculate the left and right sums for each subdivision:
Subdivision 1:
Left endpoint: t=0
Right endpoint: t=2
Velocity at the left endpoint: v(0) = 2(0)^2 + 5 = 5 m/sec
Velocity at the right endpoint: v(2) = 2(2)^2 + 5 = 17 m/sec
Subdivision 2:
Left endpoint: t=2
Right endpoint: t=4
Velocity at the left endpoint: v(2) = 17 m/sec
Velocity at the right endpoint: v(4) = 2(4)^2 + 5 = 37 m/sec
Subdivision 3:
Left endpoint: t=4
Right endpoint: t=6
Velocity at the left endpoint: v(4) = 37 m/sec
Velocity at the right endpoint: v(6) = 2(6)^2 + 5 = 77 m/sec
Subdivision 4:
Left endpoint: t=6
Right endpoint: t=8
Velocity at the left endpoint: v(6) = 77 m/sec
Velocity at the right endpoint: v(8) = 2(8)^2 + 5 = 133 m/sec
Step 3: Calculate the distance traveled during each subdivision.
To calculate the distance traveled during each subdivision, we multiply the average velocity by the width of the subdivision.
Distance in subdivision 1: (average velocity) * (width) = ((5+17)/2) * 2 = 11 * 2 = 22 meters
Distance in subdivision 2: ((17+37)/2) * 2 = 27 * 2 = 54 meters
Distance in subdivision 3: ((37+77)/2) * 2 = 57 * 2 = 114 meters
Distance in subdivision 4: ((77+133)/2) * 2 = 105 * 2 = 210 meters
Step 4: Calculate the total distance traveled.
To calculate the total distance traveled, we simply add up the distances from each subdivision.
Total distance traveled = Distance in subdivision 1 + Distance in subdivision 2 + Distance in subdivision 3 + Distance in subdivision 4
Total distance traveled = 22 + 54 + 114 + 210 = 400 meters
Therefore, the estimated distance traveled between t=0 and t=8, using the average of the left and right sums with 4 subdivisions, is 400 meters.