Answer
a) $87$ inches
b) $87$ inches
Work Step by Step
a. Using left-endpoint values:
Let's calculate the estimated distance using left-endpoint values:
For the subinterval [0, 1], velocity = 0
For the subinterval [1, 2], velocity = 12
For the subinterval [2, 3], velocity = 22
For the subinterval [3, 4], velocity = 10
For the subinterval [4, 5], velocity = 5
For the subinterval [5, 6], velocity = 13
For the subinterval [6, 7], velocity = 11
For the subinterval [7, 8], velocity = 6
For the subinterval [8, 9], velocity = 2
For the subinterval [9, 10], velocity = 6
Now, let's calculate the areas of the rectangles and sum them up:
Area = (1) $\times$ 0 + (1) $\times$ 12 + (1) $\times$ 22 + (1) $\times$ 10 + (1) $\times$ 5 + (1) $\times$ 13 + (1) $\times$ 11 + (1) $\times$ 6 + (1) *$\times$2 + (1) $\times$ 6
= 0 + 12 + 22 + 10 + 5 + 13 + 11 + 6 + 2 + 6
= 87 in.
Therefore, the estimated distance traveled by the model train engine using left-endpoint values is 87 inches.
b. Using right-endpoint values:
Let's calculate the estimated distance using right-endpoint values:
For the subinterval [0, 1], velocity = 12
For the subinterval [1, 2], velocity = 22
For the subinterval [2, 3], velocity = 10
For the subinterval [3, 4], velocity = 5
For the subinterval [4, 5], velocity = 13
For the subinterval [5, 6], velocity = 11
For the subinterval [6, 7], velocity = 6
For the subinterval [7, 8], velocity = 2
For the subinterval [8, 9], velocity = 6
For the subinterval [9, 10], velocity = 0
Now, let's calculate the areas of the rectangles and sum them up:
Area = (1) $\times$ 12 + (1) $\times$ 22 + (1) $\times$ 10 + (1) $\times$ 5 + (1) $\times$ 13 + (1) $\times$ 11 + (1) $\times$ 6 + (1) $\times$ 2 + (1) $\times$ 6 + (1) $\times$ 0
= 12 + 22 + 10 + 5 + 13 + 11 + 6 + 2 + 6 + 0
= 87 in.
Therefore, the estimated distance traveled by the model train engine using right-endpoint values is also 87 inches.
