Applications of Integration – Study Notes for Exam Preparation

Applications of Integration

Finding Areas Between Curves

One of the fundamental applications of integration is to determine the area between two curves. This involves integrating the difference between the upper and lower functions over a specified interval.

• Key Point 1: If $y = f(x)$ is above $y = g(x)$ on $[a, b]$, the area between the curves is given by:

$A = \int_a^b [f(x) - g(x)]\,dx$

• Key Point 2: Sketching the region helps to identify the limits of integration and which function is on top.

• Example: Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$.

$A = \int_0^1 (x - x^2)\,dx = \left[\frac{1}{2}x^2 - \frac{1}{3}x^3\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Volumes of Solids of Revolution

Integration can be used to find the volume of a solid formed by revolving a region around an axis. The two main methods are the disk/washer method and the shell method.

• Disk/Washer Method: Used when slicing perpendicular to the axis of revolution.

$V = \pi \int_a^b [R(x)^2 - r(x)^2] dx$

• Shell Method: Used when slicing parallel to the axis of revolution.

$V = 2\pi \int_a^b (\text{radius})(\text{height}) dx$

• Example: Find the volume of the solid formed by revolving $y = x^2$ about the x-axis from $x = 0$ to $x = 1$.

$V = \pi \int_0^1 (x^2)^2 dx = \pi \int_0^1 x^4 dx = \pi \left[\frac{1}{5}x^5\right]_0^1 = \frac{\pi}{5}$

Work Applications

Integration is used to calculate work done in various physical contexts, such as pumping water or lifting objects with variable force.

• Key Point 1: Work is the integral of force over distance.

$W = \int_a^b F(x) dx$

• Example: Calculating the work required to pump water out of a tank.

Additional info: The force may vary with position, and the limits of integration depend on the geometry of the problem.

Practice Exam Problems

The following are representative problems for exam preparation:

• Find the area between two curves.

• Compute the volume of a solid of revolution using the disk/washer or shell method.

• Set up and evaluate integrals for work done in pumping fluids or lifting objects.

Sample Table: Comparison of Methods for Volumes of Revolution

Method	When to Use	Formula
Disk/Washer	Perpendicular slices to axis	$V = \pi \int_a^b [R(x)^2 - r(x)^2] dx$
Shell	Parallel slices to axis	$V = 2\pi \int_a^b (\text{radius})(\text{height}) dx$

Additional info: For all applications, carefully sketch the region, identify the limits of integration, and choose the appropriate method based on the problem's geometry.