Lecture Note: §6 Applications of Definite Integrals

Last Update: 2025-11-20

In this chapter, we will see some of the many additional applications of definite integrals. We will use the definite integral to define and find volumes, lengths of plane curves, and areas of surfaces of revolution.

§6.1 Volumes Using Cross-Sections

• Cross-Section Methods (Slicing by Parallel Planes):
  • The volume of a solid of integrable cross-sectional area \(A(x)\): \(V = \int_{a}^{b} A(x)\, dx\)

• Solids of Revolution

  • The Disk Method
  - \(V = \int_{a}^{b} A(x)\, dx = \int_{a}^{b} \pi \left[R(x)\right]^{2}\, dx\) - \(V = \int_{c}^{d} A(y)\, dy = \int_{c}^{d} \pi \left[R(y)\right]^{2}\, dy\)
  • The Washer Method
  - \(V = \int_{a}^{b} A(x)\, dx = \int_{a}^{b} \pi \left(\left[R(x)\right]^{2}- \left[r(x)\right]^{2}\right)\, dx\) - \(V = \int_{c}^{d} A(y)\, dy = \int_{c}^{d} \pi \left(\left[R(y)\right]^{2}- \left[r(y)\right]^{2}\right)\, dy\)

§6.2 Volumes Using Cylindrical Shells

• The Shell Method (Slicing by Cylindrical Shells):

  • \(V = \int_{a}^{b} 2\pi (\text{shell radius})(\text{shell height})\, dx\)

  • \(V = \int_{c}^{d} 2\pi (\text{shell radius})(\text{shell height})\, dy\)

• Disks and Washers versus Cylindrical Shells.

§6.3 Arc Length

• The length (arc length) of the curve \(y = f(x)\)

  • \(L = \int_{a}^{b} \sqrt{1 + [f’(x)]^{2}}\, dx = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\, dx\)

• The Differential Formula for Arc Length

  • The arc length function: \(s(x) = \int_{a}^{x} \sqrt{1+ [f\prime(t)]^{2}} dt\)

  • The differential of arc length

  - \(ds = \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx\) or \(ds = \sqrt{dx^{2} + dy^{2}}\)

§6.4 Areas of Surfaces of Revolution

• Surface Area for Revolution

  • \(S=\int_{a}^{b} 2\pi y\,d s = \int_{a}^{b} 2\pi y\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}\,d x=\int_{a}^{b}2\pi f(x)\sqrt{1+(f^{\prime}(x))^{2}}\,d x\)

  • The general version: change \(y\) to the distance of the surface to the revolving axis.