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Section 8.1 The Nature of Integral Problems
and Ways to Approach Them

Think "integral" any time that you are confronted with a situation where you know how fast Quantity A varies with respect to Quantity B and you want to know an amount of Quantity A that has accumulated with respect to Quantity B.

Example 1. A rock is thrown vertically from a height of 2 meters with an initial velocity of 21 m/s. Acceleration due to gravity is -9.8 (m/s)/s. What is the rock's height after 2.8 seconds? What was the rock's height at each number of seconds after being thrown?

As was the case in Section 7.3 (Optimization), the most time-consuming part of answering such questions is to construct an appropriate model of relationships among quantities. Once you have a rate of change function that models how fast one quantity varies with respect to another, then all that remains is to integrate the rate of change function over an appropriate interval.

It is worthwhile to note something about the problem in Example 1. Contrary to intuition, you must, in principle, answer the general question before you can answer the specific question. You must answer the question of the rock's height at each moment in time to answer the question about its height after 2.8 seconds have elapsed.

Solution to Example 1. First, notice that the two relevant quantities in Example 1 are the rock's velocity at each moment in time and the elapsed time since being thrown. Then we can reason as follows:

We could have defined added velocity as an integral, since it builds up as t varies (see Figure 8.1.2, right).

 
Figures 8.1.1 and 8.1.2. Solutions to Example 1, entered into GC.

In both solutions to Example 1, we defined the distance function in open form, as an integral. Defining an accumulation function in open form is always good. It outlines our solutions conceptually.

However, some questions can be difficult to answer precisely based on the open-form-integral approach:

After how many seconds does the rock hit the ground?

We could approximate an answer by using GC's tracing feature on the graph of $y=s(x)$ to see that the rock hits the ground approximately 4.379 seconds after being released.

To answer this question precisely, we need to define $s$ symbolically. Using the FTC, we see that
$$\begin{align}s(t)&=2+\int_0^t v(u)du\\[1ex]
&=2+\int_0^t (21-9.8u)du\\[1ex]
&=2+\left. \left(21u-\frac{9.8}{2}u^2\right)\right|_{u=0}^{u=t}\\[1ex]
&=2+21t-\frac{9.8}{2}t^2.\end{align}$$
Using the quadratic formula to solve for t in $$2+21t-\frac{9.8}{2}t^2=0,$$we get $$t=\frac{-21-\sqrt{21^2-4\cdot \frac{-9.8}{2} \cdot 2}}{2 \cdot \frac{-9.8}{2}}\approx 4.37911.$$

According to our model, the ball hit the ground approximately 4.37911 seconds after being thrown.

Reflection 8.1.1. Do the graphs in Figures 8.1.1 and 8.1.2 show the ball's path after being thrown? If not, what do they show?

Two Modes of Approaching Integral Problems

The discussion in Example 1 illustrated strengths and weaknesses of two modes of approaching problems that involve integrals.

In Chapter 8 we will concentrate on the first mode of approaching problems. Modeling a situation using integrals allows us to focus on the conceptual nature of the situation's quantities and relationships among them.

In Chapter 9 (Integration Techniques), we will concentrate on the second mode, using the FTC to find closed form definitions of the same functions that we defined in Chapter 8 using integrals. Finding closed form definitions of integral functions will allow us to investigate structural properties of situations.

Exercise Set 8.1

  1. Download the GC file for Section 8.1 Example 1 (a rock thrown vertically from a height of 2 meters with initial velocity of 21 m/s). Use the functions in this file to:
    1. Give a numerical estimate for s(1.9):
    2. Give a numerical estimate for v(1.9):
    3. Represent s(1.9) in terms of the definition of $s$:
    4. Represent v(1.9) in terms of the definition of $v$:
    5. What is the quantitative meaning of s(1.9)?
    6. What is the quantitative meaning of v(1.9)?
  2. Download the GC file for Section 8.1 Example 1. Explain the meaning of the following expressions:

    1. $du$
    2. $-9.8⋅du$
    3. $v(u)⋅du$
    4. $\displaystyle{\int_0^4 -9.8⋅du}$
    5. $\displaystyle{\int_0^3 v(u)⋅du}$
    6. $\displaystyle{21 + \int_0^4 -9.8⋅du}$
    7. $\displaystyle{2 + \int_0^3 v(u)⋅du}$
  3. $V(T)$ is the volume (in $\text{cm}^3$) of a container at the moment the temperature is T degrees Celsius. $H(T)$ is the rate of change of volume with respect to temperature at the moment the temperature is T degrees C.
    1. $V(T)$ can be represented as the _____ of $H(\_\_\_\_)$ with respect to _____.
    2. Represent the small bit of volume that accumulates as the temperature varies slightly from 56 degrees C:
    3. Use $V(T)$ to represent the exact net accumulation of volume as the temperature varies from 56 degrees C to 51 degrees C.
    4. Suppose that the container's volume was 1023 $\text{cm}^3$ when the temperature was 51 degrees C.
      1. Represent the container's total volume in $\text{cm}^3$ when the temperature is t degrees C.
      2. Rewrite your answer to "i" using function notation instead of the number 1023. In what way is this representation more general than the one you wrote for "i"?
  4. $q(t)$ is the mass, in grams, of a yeast culture t hours after it started growing. $r(t)$ is the rate, in $\mathrm{g/sec}$, at which the yeast mass is increasing with respect to time.
    1. $q(t)$ can be represented by the _____ of $r(\_\_\_)$ with respect to _____.
    2. Represent the small bit of mass that accumulates as elapsed time varies slightly from 0.75 hours:
    3. Use $q(t)$ to represent the exact net accumulation of mass as elapsed time varies from 1.2 hours to 2.1 hours.
    4. Use $q(t)$ to represent the exact net accumulation of mass when we imagine elapsed time varying from 3.2 hours to 2.1 hours.
    5. Suppose that the culture's mass was 102.7 grams when 1.72 hours had elapsed.
      1. Represent the culture's initial mass (when $t=0$).
      2. Represent the culture's total mass in grams after t hours have elapsed.

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