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Section 4.2 Approximating definite integrals (IN2)

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 58, sketch a labeled graph of the velocity function \(v(t) = 3\text{.}\)

Figure 58. At left, axes for plotting \(y = v(t)\text{;}\) at right, for plotting \(y = s(t)\text{.}\)

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 58, 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 59 and graph given in Figure 60.

Table 59. Velocity data for the person walking.
\(t\) \(v(t)\)
\(0.00\) \(1.500\)
\(0.25\) \(1.789\)
\(0.50\) \(1.938\)
\(0.75\) \(1.992\)
\(1.00\) \(2.000\)
\(1.25\) \(2.008\)
\(1.50\) \(2.063\)
\(1.75\) \(2.211\)
\(2.00\) \(2.500\)
Figure 60. The graph of \(y = v(t)\text{.}\)

(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.