Applications of integrals (area, volume, accumulation) Knowledge-Based Study

Study notes

1. The Net Change Theorem

   The definite integral can be interpreted as the total accumulation of a quantity over an interval when its rate of change is known. This principle is formalized by the Net Change Theorem:

   $$\int_{a}^{b} F'(t) \, dt = F(b) - F(a)$$

   This theorem states that the integral of a rate of change gives the net change in the quantity over the interval from $t = a$ to $t = b$.

   Motion Along a Line

   A classic application is linear motion, where an object moves with velocity $v(t)$:

   • Displacement (Net Change in Position): $$s(b) - s(a) = \int_{a}^{b} v(t) \, dt$$
   • Total Distance Traveled: Because velocity can be negative, total distance accounts for all movement regardless of direction: $$\text{Total Distance} = \int_{a}^{b} |v(t)| \, dt$$

   Computing Areas in the Plane

   Definite integrals can be used to find the area bounded by two or more continuous curves.

   Integration with Respect to $x$

   If $f(x)$ and $g(x)$ are continuous functions such that $f(x) \ge g(x)$ on the interval $[a, b]$, the area $A$ bounded between them is given by:

   $$A = \int_{a}^{b} [f(x) - g(x)] \, dx$$

   Here, $f(x)$ represents the upper boundary curve and $g(x)$ represents the lower boundary curve.

   Integration with Respect to $y$

   When regions are more naturally bounded by functions of $y$, integrating with respect to $y$ simplifies calculations. If $f(y) \ge g(y)$ on $[c, d]$ (meaning $x = f(y)$ lies to the right of $x = g(y)$):

   $$A = \int_{c}^{d} [f(y) - g(y)] \, dy$$

   If curves intersect multiple times, partition the interval into subintervals where the relative order of the curves does not change, or use the absolute value expression $A = \int_{a}^{b} |f(x) - g(x)| \, dx$.

   The Disk and Washer Methods

   When a planar region is revolved around an axis of rotation, it sweeps out a three-dimensional solid called a solid of revolution. If the cross-sections perpendicular to the axis of revolution are circular disks or washers, their areas can be integrated to yield the total volume.

   The Disk Method

   If the cross-section is a solid circular disk of radius $R(x)$, the area of each cross-section is $A(x) = \pi [R(x)]^2$. Revolving around a horizontal axis gives:

   $$V = \pi \int_{a}^{b} [R(x)]^2 \, dx$$

   The Washer Method

   If the region rotated does not touch the axis of rotation, the cross-section is an annulus (washer) with outer radius $R(x)$ and inner radius $r(x)$. The cross-sectional area is $A(x) = \pi \left([R(x)]^2 - [r(x)]^2\right)$, leading to:

   $$V = \pi \int_{a}^{b} \left( [R(x)]^2 - [r(x)]^2 \right) \, dx$$

   For rotation about a vertical axis, the variable of integration shifts from $x$ to $y$, using horizontal slices of thickness $dy$.

   The Cylindrical Shell Method

   The washer method requires slicing perpendicular to the axis of rotation. In contrast, the method of cylindrical shells integrates concentric thin cylindrical shells parallel to the axis of rotation.

   Formulation

   Consider a region bounded by $y = f(x)$, $y = 0$, from $x = a$ to $x = b$ (where $0 \le a < b$), revolved around the $y$-axis. Approximating the solid with nested cylindrical shells of radius $x$, height $f(x)$, and thickness $dx$ gives the surface area $2\pi x f(x)$. Integrating over all shells yields:

   $$V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx = 2\pi \int_{a}^{b} x \, f(x) \, dx$$

   Choosing Between Washers and Shells

   • Washer Method: Slices are perpendicular to the axis of revolution.
   • Shell Method: Slices are parallel to the axis of revolution.

   The shell method is particularly advantageous when expressing $x$ in terms of $y$ is algebraically difficult, or when the washer method would require breaking the region into multiple separate integrals.