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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 you have 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).
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 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 thatAccording 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?
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.
Download the GC file for Section 8.1 Example 1. Explain the meaning of the following expressions:
$du$
$-9.8⋅du$
$v(u)⋅du$
$\displaystyle{\int_0^4 -9.8⋅du}$
$\displaystyle{\int_0^3 v(u)⋅du}$
$\displaystyle{21 + \int_0^4 -9.8⋅du}$
$V(T)$ can be modeled with the _____ of $H(T)$ with respect to _____.
Represent the small bit of volume that accumulates as the temperature changes slightly from 56 degrees C:
Use $H(T)$ to represent the exact net accumulation of volume as the temperature changes from 56 degrees C to 51 degrees C.
Use $V(T)$ to represent the exact net accumulation of volume as the temperature changes from 56 degrees C to 51 degrees C.
Suppose that the container's volume was 1023 $\text{cm}^3$ when the temperature was 51 degrees C.
$q(t)$ can be modeled with the _____ of $r(t)$ with respect to _____.
Represent the small bit of mass that accumulates as elapsed time changes slightly from 0.75 hours:
Use $q(t)$ to represent the exact net accumulation of mass as elapsed time changes from 1.2 hours to 2.1 hours.
Use $q(t)$ to represent the exact net accumulation of mass when we imagine elapsed time changing from 3.2 hours to 2.1 hours.
Suppose that the culture's mass was 102.7 grams when 1.72 hours had elapsed.