Advanced Integration Techniques and the Net Change Theorem

Cylindrical Shells and Geometric Orientation

In the context of the cylindrical shells method for volumes of revolution, the distance is determined by subtracting the bound of the function, such as $y^2 + 1$, from the outer bound. When a cylinder is positioned on its side, the standard definitions of height and radius are rotated. In this engineering application, the height term refers to the horizontal or sideways distance, while the radius term refers to the vertical distance. For instance, if a function was being revolved around the line $x = 1$, the radius would be represented as a function of $y$, potentially appearing as $y - 1$.

Definitions and Properties of Logarithms and Exponentials

The natural logarithm, denoted as $\ln(x)$, is defined as the logarithm with base $e$, specifically $\log_e(x)$. This is distinct from common logarithms, such as $\log_{10}$, which uses base 10. A fundamental property of these functions is their inverse relationship; for example, raising 10 to the power of $\log_{10}(x)$ results in $x$, just as raising $e$ to the natural log of $x$ results in $x$.

Algebraic Solving Methods for Transcendental Equations

When solving equations involving natural logarithms, such as $\ln((x+1)^2) = 5$, a primary method is to raise $e$ to the power of both sides of the equation. This utilizes the inverse property to cancel the natural log on the left side, leaving $(x+1)^2 = e^5$. Solving for $x$ requires taking the square root, resulting in $x+1 = \pm \sqrt{e^5}$, and then subtracting 1 to find the candidate solutions: $x = -1 \pm e^{2.5}$.

It is critical to verify if candidate solutions are actually defined within the original equation. Because the argument of a logarithm must be positive, solutions must be checked against the original expression. In the case of $\ln((x+1)^2)$, the squared term $(x+1)^2$ ensures the argument is always positive as long as $x \neq -1$. Therefore, both candidate solutions are valid. If a substitution resulted in a negative number inside a non-squared natural log, such as $\ln(-10)$, that solution would be discarded.

In cases such as $e^4 x = 0$, since $e^4$ is a non-zero constant, the equation can be simplified by dividing both sides by $e^4$, resulting in $x = 0$. However, if an equation is presented such that after taking the natural log of both sides, one ends up with the natural log of zero ($\ln(0)$), it indicates that the specific method may not work or that a solution does not exist, as $\ln(0)$ is undefined.

Another example of simplification involves the equation $\ln(x-1) = \log_2(e)$. By exponentiating both sides with base $e$, the left side simplifies to $x-1$, leading to the solution $x = 1 + e^{\log_2(e)}$. To check this, substituting it back into the natural log gives $\ln(e^{\log_2(e)})$, which is $\log_2(e)$. Since the exponent of $e$ is always positive, the natural log remains defined.

Integration of Multi-Section Transcendental Functions

Complex problems often combine concepts from different sections, such as finding the area between curves (Section 5.1) and performing calculus with exponential functions (Section 6.2 and 6.3). Consider finding the area between the function $f(x) = \sin(x) e^{\cos(x)}$ and the horizontal line $y = 0$ over the interval from $0$ to $\pi$.

To set up this integral, one must determine which function is on top. Since $e^{\text{anything}}$ is always positive, the sign of the entire function depends on $\sin(x)$. On the interval $[0, \pi]$, $\sin(x)$ is positive, meaning $f(x)$ is above $y=0$. The integral is set up as:

$\int_0^{\pi} \sin(x) e^{\cos(x)} \, dx$

To solve this, a $u$-substitution is used. Let $u = \cos(x)$, which implies that the derivative $du = -\sin(x) \, dx$, or $-du = \sin(x) \, dx$.

There are two valid methods for handling the bounds of a definite integral during substitution:

1. Change the bounds of integration to match the variable $u$. For the upper bound, $\cos(\pi) = -1$. For the lower bound, $\cos(0) = 1$. The integral becomes $\int_1^{-1} -e^u \, du$, which is equivalent to $\int_{-1}^1 e^u \, du$.
2. Solve the antiderivative in terms of $u$, substitute the original variable $x$ back in, and then use the original bounds of $0$ and $\pi$.

The antiderivative of $-e^u$ is $-e^u$. If using the first method, the evaluation is $[-e^u]_1^{-1} = -e^{-1} - (-e^1) = e - \frac{1}{e}$. Since this represents an area, the final result must be positive.

The Net Change Theorem and Physics Applications

The Net Change Theorem states that the integral of a rate of change is the net change in the quantity. For instance, if $a(t)$ is the acceleration of a particle at time $t$, the integral of acceleration over a time interval $[0, 1]$ represents the change in velocity:

$\int_0^1 a(t) \, dt = v(1) - v(0)$

This represents the increase in velocity between zero and one seconds. If the result is negative, it indicates a decrease. To find the total velocity at time $t = 1$, one must add the initial velocity $v(0)$ to the integral of acceleration. If $v(0) = 1\,m/s$ and the integral yields $15\,m/s$, the total velocity $v(1)$ is $16\,m/s$.

In the context of position and velocity, the integral of velocity $\int_{t_1}^{t_2} v(t) \, dt$ yields the displacement, which is the net increase or change in position. This is distinct from total distance traveled. Velocity and displacement are oriented quantities (vectors), whereas speed and distance are magnitudes (scalars). Speed is the absolute value of velocity.

Applied Modeling: Bee Population Change

In a population model where $P(t)$ represents the rate of change of a bee population $t$ years after the year 2026, the net change in population between 2026 and 2030 is found by integrating the rate of change over that interval:

$\text{Change in Population} = \int_0^4 P(t) \, dt$

To calculate the total population of bees in 2030, the initial population from 2026 must be added to this integral. If there were 100 bees to start with, the formula is:

$\text{Total Population} = 100 + \int_0^4 P(t) \, dt$

Note that the initial constant (100) must be added outside the integral. If it were added inside the integral, it would be subject to integration (e.g., $\int_0^4 (P(t) + 100) \, dt$), which would incorrectly multiply the constant by the length of the interval (resulting in an addition of 400 instead of 100).

Inverse Functions

The inverse of the natural logarithm function $\ln(x)$ is the exponential function $e^x$. Understanding this relationship is essential for both solving algebraic equations and performing calculus involving logarithmic and exponential terms.

Questions & Discussion

Question: How does log tie into natural log, and how do we handle base 10? Answer: Natural log is specifically $\log_e$. Base 10 logs work the same way algebraically; for example, $10^{\log_{10}(x)} = x$.

Question: Do you always have to change the bounds when doing $u$-substitution for a definite integral? Answer: No, you can either change the bounds to match $u$ or find the antiderivative, plug the original $x$ variable back in, and use the original bounds. Both methods are valid as long as you do not mix $u$ bounds with $x$ variables or vice versa.

Question: Can you explain the difference between displacement and distance in the context of the Net Change Theorem? Answer: Displacement is the integral of velocity and is oriented (can be positive or negative, representing the net change in position). Distance is the integral of the absolute value of velocity (speed) and represents the total path traveled regardless of direction.

Question: For the bee population, if we want to know total population, can we just pull the constant out? Answer: You pull out constants only when they are being multiplied by the function. When you are adding an initial value to a change, that constant is added to the result of the integral.