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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 changes 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 one has a rate of change function that models how fast one quantity changes 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, in terms of an integral. This is actually a good thing to do. A computer doesn't care in which form we define a function as long as it can compute a value. Moreover, it helps us to outline our solutions conceptually.

However, an additional question causes a problem for 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.

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 (Figure 8.1.2) HERE.
      1. Give a real number estimate for s(7.2):
      2. Give a real number estimate for v(1.9):
      3. Represent the exact value of s(7.2):
      4. Represent the exact value of v(1.9):
      5. What is the quantitative meaning of s(7.2)?
      6. What is the quantitative meaning of v(1.9)?
  2. Essay Question: Download the GC file for Section 8.1 Example 1 (Figure 8.1.2) HERE. 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. $A(T)$ is the number of people that are attending a concert when the temperature is T degrees Celsius, in thousands of people. $F(T)$ is the rate at which the attendance grows with respect to temperature, in thousands of people per degree Celsius.
    1. $A(T)$ can be modeled with the _____ of $F(T)$ with respect to _____.

    2. Represent the small bit of attendance that accumulates as the temperature changes slightly from 56 degrees C:

    3. Use $F(T)$ to represent the exact net accumulation of attendance as the temperature changes from 56 degrees C to 51 degrees C.

    4. Use $A(T)$ to represent the exact net accumulation of attendance as the temperature changes from 56 degrees C to 51 degrees C.

    5. Suppose that there were 4,750 people at the concert when the temperature was 51 degrees C. Use $F(T)$ to represent the total number of people in thousands attending when the temperature is $T$ degrees C.

    6. Suppose there were 4,750 people at the concert when the temperature was 51 degrees C. Use $A(T)$ to represent the total number of people in thousands attending when the temperature is T degrees C.


  4. $q(t)$ is the volume, in $\mathrm{cm^3}$, of yeast in my yeast culture t hours after it started growing. $r(t)$ is the rate, in $\mathrm{cm^3/sec}$, at which the yeast volume is increasing with respect to time.