Learning Outcomes
- Approximate definite integrals using Riemann sums.
Section 4.2 Approximating definite integrals (IN2)
Subsection 4.2.1 Activities
Activity 4.2.1.
Suppose that a person is taking a walk along a long straight path and walks at a constant rate of 3 miles per hour.
(a)
On the left-hand axes provided in Figure 82, sketch a labeled graph of the velocity function \(v(t) = 3\text{.}\)
Note that while the scale on the two sets of axes is the same, the units on the right-hand axes differ from those on the left. The right-hand axes will be used in question (d).
(b)
How far did the person travel during the two hours? How is this distance related to the area of a certain region under the graph of \(y = v(t)\text{?}\)
(c)
Find an algebraic formula, \(s(t)\text{,}\) for the position of the person at time \(t\text{,}\) assuming that \(s(0) = 0\text{.}\) Explain your thinking.
(d)
On the right-hand axes provided in Figure 82, sketch a labeled graph of the position function \(y = s(t)\text{.}\)
(e)
For what values of \(t\) is the position function \(s\) increasing? Explain why this is the case using relevant information about the velocity function \(v\text{.}\)
Activity 4.2.2.
Suppose that a person is walking in such a way that her velocity varies slightly according to the information given in Table 83 and graph given in Figure 84.
| \(t\) | \(v(t)\) |
| \(0.00\) | \(1.500\) |
| \(0.25\) | \(1.789\) |
| \(0.50\) | \(1.938\) |
| \(0.75\) | \(1.992\) |
| \(1.00\) | \(2.000\) |
| \(1.25\) | \(2.008\) |
| \(1.50\) | \(2.063\) |
| \(1.75\) | \(2.211\) |
| \(2.00\) | \(2.500\) |
(a)
Using the grid, graph, and given data appropriately, estimate the distance traveled by the walker during the two hour interval from \(t = 0\) to \(t = 2\text{.}\) You should use time intervals of width \(\Delta t = 0.5\text{,}\) choosing a way to use the function consistently to determine the height of each rectangle in order to approximate distance traveled.
(b)
How could you get a better approximation of the distance traveled on \([0,2]\text{?}\) Explain, and then find this new estimate.
(c)
Now suppose that you know that \(v\) is given by \(v(t) = 0.5t^3-1.5t^2+1.5t+1.5\text{.}\) Remember that \(v\) is the derivative of the walker's position function, \(s\text{.}\) Find a formula for \(s\) so that \(s' = v\text{.}\)
(d)
Based on your work in (c), what is the value of \(s(2) - s(0)\text{?}\) What is the meaning of this quantity?
Activity 4.2.3.
Explain how to approximate the area under the curve \(f(x)=-9 \, x^{3} + 3 \, x - 9\) on the interval \([4,10]\) using a right Riemann sum with 3 rectangles of uniform width.
