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Section 5.4 Skills: Work

Section 5.4: Work

This section covers calculating work done by constant and variable forces, with applications to springs, cables, and tank pumping problems.

Learning Path

graph LR
    subgraph Foundation
        A["Work with<br/>Integrals"]
    end

    subgraph Applications
        B["Spring<br/>Work"]
        C["Pumping<br/>Work"]
    end

    A --> B
    A --> C
    B --> C

    style A fill:#d1fae5,stroke:#a565f0,stroke-width:3px
    style B fill:#d1fae5,stroke:#a565f0,stroke-width:2px
    style C fill:#d1fae5,stroke:#a565f0,stroke-width:2px

    click A "work-with-integrals.html"
    click B "spring-work.html"
    click C "pumping-work.html"

Skills in This Section

Skill	Description	Difficulty	Time
Work with Integrals	Foundation: $W = Fd$ and $W = \int f(x)\,dx$	Intermediate	~20 min
Spring Work	Hooke's Law: $f(x) = kx$ and spring applications	Intermediate	~20 min
Pumping Work	Cables, chains, and tank pumping problems	Advanced	~25 min

Key Formulas

Formula	Application
$W = Fd$	Constant force over distance $d$
$W = \int_a^b f(x)\,dx$	Variable force from $x=a$ to $x=b$
$f(x) = kx$	Hooke's Law for springs
$W = \frac{1}{2}k(b^2 - a^2)$	Spring work from $x=a$ to $x=b$
$W = \int_a^b \rho g \cdot A(x) \cdot d(x)\,dx$	Pumping work (fluid in tanks)

Prerequisites

Before starting this section, ensure you're comfortable with:

Definite integrals and the Fundamental Theorem of Calculus
Riemann sums (conceptual understanding of how integrals arise)
Basic polynomial integration (power rule for $\int x^n\,dx$)

Tip: Each skill page has a "Before You Start" self-check. If you struggle with those diagnostics, follow the links to review prerequisites before proceeding.

Recommended Study Order

Start with Work with Integrals: establishes the foundation
Then Spring Work: applies the integral to Hooke's Law
Finally Pumping Work: extends to multi-dimensional problems

← Section 5.3 Skills	Section 5.4 Overview	Section 5.5 Skills →

Last updated: 2026-01-23