Calculus II: Integration, Volumes, and Work Applications

• $\int \sec(x) \, dx = \ln |\sec(x) + \tan(x)| + C$
• $\int \csc(x) \, dx = \ln |\csc(x) - \cot(x)| + C$

Volumes of Solids of Revolution

One of the most common applications of the definite integral is finding the volume of a solid created by revolving a 2D region around an axis. The method you choose depends entirely on the axis of revolution and the orientation of your functions.

The Disk and Washer Method

When the region is revolved around the $x$-axis (or a horizontal line) and the cross-sections are perpendicular to that axis, we use the Disk/Washer method. If the region is flush against the axis, it is a disk; if there is a gap, it is a washer. The formula for a function $y = f(x)$ revolved around the $x$-axis is: $V = \pi \int_{a}^{b} [f(x)]^2 \, dx$

The Shell Method

If you are revolving a region around a vertical line (such as $x = -2$), the Shell Method is often much more efficient because it avoids the need to algebraically invert your function from $y=f(x)$ to $x=g(y)$. The volume is calculated by summing the surface areas of infinitely many thin cylindrical shells: $V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx$

Physical Applications: Work and the Uniform-Density Chain

In physics, Work ($W$) is defined as the integral of force over a distance. While constant force is easy to calculate ($W = F \cdot d$), calculus is required when the force changes as the object moves.

The Variable Force of a Hanging Chain

A classic exam problem involves pulling a uniform-density chain up a building. As you pull the chain up, the amount of chain hanging over the edge decreases, which means the weight (the force you must exert) also decreases.

To solve these problems, follow these steps:

1. Determine Linear Density ($\lambda$): Find the mass per unit length ($\lambda = \frac{\text{total mass}}{\text{total length}}$).
2. Express Force as a Function of Position: The force $F(x)$ at any point $x$ is the weight of the remaining hanging segment. Since weight is $\text{mass} \times g$, and mass is $\text{density} \times \text{length}$, your force function will depend on how much chain is left.
3. Integrate: Set up the integral of the force function over the distance moved: $W = \int_{a}^{b} F(x) \, dx$

Example Scenario: If a 160m chain with a mass of 80kg is hanging from a building, and you need to pull the first 10m up, your integral will account for the changing weight of the remaining 150m to 160m of chain as it moves toward the top.

Critical Reminder: Don't forget to include the acceleration due to gravity ($g = 9.8 \, \text{m/s}^2$) when converting mass to weight (force).