Step 1 :Divide the interval \(1 \leq t \leq 4\) into \(n=500\) subdivisions.
Step 2 :Calculate the width of each subdivision: \(\Delta t = \frac{4-1}{500} = \frac{3}{500}\)
Step 3 :Calculate the upper and lower estimates of the distance traveled:
Step 4 :For the upper estimate, calculate the maximum value of \(f(t)\) in each subdivision and multiply it by the width of the subdivision.
Step 5 :For the lower estimate, calculate the minimum value of \(f(t)\) in each subdivision and multiply it by the width of the subdivision.
Step 6 :Calculate the difference between the upper and lower estimates:
Step 7 :Difference = Upper estimate - Lower estimate
Step 8 :Calculate the upper estimate:
Step 9 :Upper estimate = \(\Delta t \cdot (f(1) + f(1+\Delta t) + f(1+2\Delta t) + ... + f(4-\Delta t) + f(4))\)
Step 10 :Calculate the lower estimate:
Step 11 :Lower estimate = \(\Delta t \cdot (f(1) + f(1+\Delta t) + f(1+2\Delta t) + ... + f(4-\Delta t) + f(4))\)
Step 12 :Calculate the difference between the upper and lower estimates:
Step 13 :Difference = Upper estimate - Lower estimate