Exploring Accumulations of Change - AP Calculus AB Study Guide

The Core Idea: Exploring Accumulations of Change

The fundamental concept of this topic is that the definite integral is a tool for calculating the total accumulation or net change of a quantity over a specific interval. If we know the rate at which a quantity is changing (for example, the velocity of a particle in meters per second), we can use a definite integral to find the total net change in that quantity (the particle's displacement in meters) over a period of time.

The definite integral of a rate function, $f^{\prime }(x)$, from a starting point $a$ to an ending point $b$ precisely measures the accumulation of all the instantaneous changes that occurred between $a$ and $b$. This process gives us the net change in the original quantity, $f(x)$, across that interval. This idea connects the concept of a rate of change (a derivative) to the concept of net change (a definite integral).

Key Concepts and Notation

The primary relationship in this topic connects a rate of change to a net change using a definite integral.

The Net Change Integral

If $R(t)$ is a function representing the rate of change of a quantity, then the net change of that quantity from time $t=a$ to $t=b$ is given by the definite integral:

$$\text{Net Change}={\int }_a^{b}R(t)dt$$

Units of a Definite Integral

The units of a definite integral are the product of the units of the function being integrated (the $y$-axis units) and the units of the variable of integration (the $x$-axis units).

• If $R(t)$ is measured in gallons per hour and $t$ is measured in hours, then the units of ${\int }_a^{b}R(t)dt$ are (gallons per hour) $\times$ (hours) = gallons.

• If $v(t)$ is measured in meters per second and $t$ is measured in seconds, then the units of ${\int }_a^{b}v(t)dt$ are (meters per second) $\times$ (seconds) = meters.

Understanding Net Change

A crucial nuance of the definite integral is that it calculates the net change, not necessarily the total amount of a quantity or the total distance traveled. The definite integral accumulates values, and if the rate function is negative on some intervals, that will contribute negatively to the final accumulated value.

Consider a particle's velocity, $v(t)$.

• When $v(t)>0$, the particle is moving in the positive direction, and the integral accumulates positive change (positive displacement).

• When $v(t)<0$, the particle is moving in the negative direction, and the integral accumulates negative change (negative displacement).

The definite integral ${\int }_a^{b}v(t)dt$ sums these positive and negative changes to find the particle's displacement, which is the net change in its position from time $a$ to time $b$. This is different from the total distance the particle traveled. The integral gives the final position relative to the starting position, not the length of the path taken.

Core Concepts & Rules

• The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.

• A definite integral can be interpreted as the accumulation of a function's values over an interval.

• If $f^{\prime }(x)$ is the rate of change of $f(x)$, then ${\int }_a^b{f}^{\prime }(x)dx$ represents the net change in $f(x)$ from $x=a$ to $x=b$.

• The units of the value of a definite integral ${\int }_a^{b}f(x)dx$ are the product of the units of $f(x)$ and the units of $x$.

Step-by-Step Example 1: Basic Application

Problem: A honeybee population in a hive is changing at a rate modeled by the function $P^{\prime }(t)$, measured in hundreds of bees per week, where $t$ is the number of weeks since the start of the season. Write a definite integral that represents the net change in the honeybee population from the beginning of week 2 to the end of week 10, and determine the units of your answer.

Step 1: Identify the Rate Function and the Interval

• The rate of change is given by the function $P^{\prime }(t)$.

• The interval is from $t=2$ weeks to $t=10$ weeks.

Step 2: Set Up the Definite Integral

The net change is the definite integral of the rate of change over the specified interval.

$$\text{Net Change}={\int }_2^{10}{P}^{\prime }(t)dt$$

Step 3: Interpret the Meaning and Determine the Units

• Interpretation: The expression ${\int }_2^{10}{P}^{\prime }(t)dt$ represents the net change in the number of honeybees in the hive from week 2 to week 10.

• Units: The units of $P^{\prime }(t)$ are "hundreds of bees per week." The units of $t$ are "weeks." Therefore, the units of the integral are:

$$(\text{hundreds of bees per week})\times (\text{weeks})=\text{hundreds of bees}$$

Step-by-Step Example 2: Exam-Style Application

Problem: Oil is leaking from a pipeline at a rate modeled by the differentiable function $L(t)$, where $L(t)$ is measured in gallons per hour and $t$ is measured in hours for $0\le t\le 24$.

(a) Write a definite integral that gives the total amount of oil, in gallons, that has leaked out of the pipeline from $t=4$ hours to $t=12$ hours.

(b) In the context of this problem, explain the meaning of ${\int }_0^{24}L(t)dt$.

Solution (a):

The function $L(t)$ represents the rate of leakage. To find the total amount leaked over an interval, we must integrate this rate over that interval.

• Rate Function:$L(t)$ gallons per hour

• Interval: from $t=4$ to $t=12$ hours

The definite integral is:

$${\int }_4^{12}L(t)dt$$

This integral represents the net change in the amount of oil that has leaked out, which in this context is the total amount of oil leaked during that time period.

Solution (b):

The expression ${\int }_0^{24}L(t)dt$ represents the accumulation of the leakage rate over the first 24 hours.

• Interpretation:${\int }_0^{24}L(t)dt$ is the total amount of oil, in gallons, that leaked from the pipeline during the 24-hour period from $t=0$ to $t=24$.

Using Your Calculator

This topic is primarily conceptual, focusing on setting up and interpreting integrals rather than calculating them. The setup and interpretation are analytical skills that do not require a calculator.

However, if you are given an explicit function for the rate and asked to find the value of the net change, you would use a calculator's numerical integration feature. For example, if the rate of leakage in Example 2 was given by $L(t)=15\sqrt{t}$, you could find the amount of oil leaked from $t=4$ to $t=12$ as follows.

To calculate ${\int }_4^{12}15\sqrt{t}dt$ on a TI-84 style calculator:

1. Press the math` key. 2. Select `9: fnInt(` or scroll down to it. 3. The syntax is $fnInt(expression, variable, lower bound, upper bound).

2. Enter the expression: fnInt(15√(X), X, 4, 12)

3. Press ENTER to get the numerical value of the net change.

AP Exam Quick Hit

Common Question Types

• Interpreting an Integral in Context: You will be given a rate function with units (e.g., $C^{\prime }(t)$ in dollars per year) and an integral expression (e.g., ${\int }_0^5{C}^{\prime }(t)dt$) and asked to explain its meaning. A correct response must include the quantity, the time interval, and the correct units (e.g., "the net change in cost, in dollars, from year 0 to year 5").

• Setting Up an Integral from a Scenario: You will be given a word problem describing a rate of change and asked to write, but not evaluate, a definite integral representing the net change over a specified interval.

Common Mistakes

• Missing or Incorrect Units: A common error is to omit units entirely or to state the units of the rate function (e.g., "gallons per hour") instead of the units of the accumulated quantity (e.g., "gallons").

• Vague or "Calculator-Speak" Interpretations: Avoid generic phrases like "the area under the curve." Your interpretation must be specific to the context of the problem. For example, instead of "the integral of the rate of people," write "the net change in the number of people."

• Confusing Net Change with Total Amount: The integral ${\int }_a^{b}R(t)dt$ gives the net change in a quantity, not the total amount of the quantity at time $b$. For example, if $R(t)$ is the rate of change of water in a reservoir, the integral does not give the total amount of water in the reservoir, because we do not know the initial amount.

• Ignoring the Interval: The interpretation must clearly state the interval over which the accumulation occurs (e.g., "from $t=2$ to $t=5$"). Simply saying "the change in population" is not specific enough.