Questions: Your velocity is given by v(t)=1 t^2+6 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.

Solution Steps

The interval for \( t \) is from 0 to 8 seconds. We will use 4 subdivisions to estimate the distance traveled. The width of each subdivision, \(\Delta t\), is calculated as follows:

\[ \Delta t = \frac{8 - 0}{4} = 2 \, \text{seconds} \]

For the left sum, we evaluate the velocity function at the left endpoints of each subdivision. The left endpoints are \( t = 0, 2, 4, 6 \).

\[ \begin{align_} v(0) &= 1(0)^2 + 6 = 6, \\ v(2) &= 1(2)^2 + 6 = 10, \\ v(4) &= 1(4)^2 + 6 = 22, \\ v(6) &= 1(6)^2 + 6 = 42. \end{align_} \]

The left sum is:

\[ \text{Left Sum} = \Delta t \times (v(0) + v(2) + v(4) + v(6)) = 2 \times (6 + 10 + 22 + 42) = 160 \]

For the right sum, we evaluate the velocity function at the right endpoints of each subdivision. The right endpoints are \( t = 2, 4, 6, 8 \).

\[ \begin{align_} v(2) &= 10, \\ v(4) &= 22, \\ v(6) &= 42, \\ v(8) &= 1(8)^2 + 6 = 70. \end{align_} \]

The right sum is:

\[ \text{Right Sum} = \Delta t \times (v(2) + v(4) + v(6) + v(8)) = 2 \times (10 + 22 + 42 + 70) = 288 \]

The average of the left and right sums gives an estimate of the distance traveled:

\[ s = \frac{\text{Left Sum} + \text{Right Sum}}{2} = \frac{160 + 288}{2} = 224 \]

Final Answer