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Calculus I · Unit 3B · hub

Unit 3B: Applications of Integration

Learn applications of integration through area, volume, length, mass, work, fluids, marginal quantities, probability, worked examples, visual reasoning, practice, and published exam keys.

Core textbook pathChoose the next lesson from the ordered unit map.

Core textbook

The complete Unit 3B path

Begin with one-dimensional area, build three-dimensional volume from slices and rotations, then carry the same contribution-times-width architecture into length, mass, work, fluids, economics, and probability.

What this unit teaches

Turn integrals into geometric, physical, and quantitative models.

Choose vertical or horizontal slices; model areas, solids, arc and surface length, density and balance, variable-force work, pumping, hydrostatic force, marginal totals, and probability; then defend each setup with units and geometry.

Prerequisites

Unit 3A integration foundations and a dependable derivative toolkit.

You should interpret definite integrals, find antiderivatives, use substitution when needed, solve intersections, sketch basic curves, and track units. Every Unit 3B page links back to Unit 3A when technique—not modeling—is the obstacle.

Section

Orientation and applications roadmap

Begin with a physical or geometric contribution, then let the integral add those contributions across the whole object or interval.

  1. 01Unit 3B: Applications of Integrationhub
Section

Area and volume

Let the axis and slice orientation determine every distance; top-minus-bottom, right-minus-left, radii, and shell height must all come from the same picture.

  1. 02Area Between Curveslesson
  2. 03Choosing Vertical or Horizontal Sliceslesson
  3. 04Volumes by Slicinglesson
  4. 05Disks and Washerslesson
  5. 06Cylindrical Shellslesson
  6. 07Choosing a Volume Methodlesson
Section

Length, surface, mass, and balance

Distinguish geometric size from weighted amount: arc and surface formulas stretch local distance, while density assigns unequal mass to equal pieces.

  1. 08Arc Lengthlesson
  2. 09Surface Area of Revolutionlesson
  3. 10Density and Masslesson
  4. 11Moments and Center of Masslesson
Section

Physics and quantitative applications

Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.

  1. 12Work and Variable Forcelesson
  2. 13Work Done on Springslesson
  3. 14Pumping Liquidslesson
  4. 15Fluid Pressure and Hydrostatic Forcelesson
  5. 16Recovering Totals from Marginal Quantitieslesson
  6. 17Probability Density and Expected Valuelesson
Section

Review, practice, exams, and answer keys

Mixed applications test modeling recognition: commit to a diagram and slice statement before looking for a familiar formula.

  1. 18Unit 3B Review: Applications of Integrationreview

Practice around the path

Reviews, quizzes, diagnostics, and exams

Use these after a section or whenever a worked example reveals a specific gap. The answer keys are separated so an honest first attempt stays easy.

practicePhysics Application StudioPractice Physics Application Studio with cumulative integral-calculus problems designed to test setup, computation, units, and interpretation.reviewUnit 3B Review: Applications of IntegrationReview Unit 3B Review: Applications of Integration with a structured summary, common-error repair, mixed practice, and links back to the relevant BetterGrades.reviewCommon Application Errors and How to Repair ThemIdentify and repair common integration-application errors involving slice orientation, radii, bounds, density, depth, units, and interpretation.practiceUnit 3B Cumulative PracticePractice Unit 3B Cumulative Practice with cumulative integral-calculus problems designed to test setup, computation, units, and interpretation.examUnit 3B Practice Exam ATake Unit 3B Practice Exam A, a comprehensive BetterGrades integral-calculus practice exam with a separately routed worked answer key.examUnit 3B Practice Exam BTake Unit 3B Practice Exam B, a comprehensive BetterGrades integral-calculus practice exam with a separately routed worked answer key.

Check your work

Published exam answer keys

Every exam has a separately routed, numbered key. Finish the exam first, then compare one item at a time.

Go deeper

Focused integral explorations

These articles zoom in on one modeling choice, slice geometry, or real-world interpretation. They are enrichment around the textbook path, not a replacement for it.

Focused articleAverage value of a functionA concise in-depth exploration connected to the main unit.Focused articleWasher method or shell method?A concise in-depth exploration connected to the main unit.Focused articleWork and fluid forceA concise in-depth exploration connected to the main unit.Focused articleWhen horizontal slices describe the region more cleanlyA concise in-depth exploration connected to the main unit.Focused articleA decision guide for washers, shells, and known cross-sectionsA concise in-depth exploration connected to the main unit.Focused articleA physics studio for mass, work, pumping, and pressureA concise in-depth exploration connected to the main unit.Focused articleWhy plausible integration models failA concise in-depth exploration connected to the main unit.

Repair the integration toolkit

Return easily to Unit 3A foundations

When the model is clear but the antiderivative or numerical method is not, review the matching Unit 3A technique, then return here to finish the setup, units, and interpretation.

Review Unit 3A foundations →

Unit 3B: Applications of Integration

Unit 3A built the integral as accumulated change and developed methods for computing it. Unit 3B uses that machinery in standard geometric, physical, and quantitative applications. The emphasis is not on collecting exotic formulas. It is on constructing the correct small contribution, integrating over the correct variable, and interpreting the result with units.

Unit map

• Area between curves • Volumes by slicing, disks, washers, shells, and known cross-sections • Arc length and surface area • Density, mass, moments, and center of mass • Work, springs, pumping, and fluid pressure • Accumulation from marginal quantities and probability densities • Review, modeling practice, practice exams and published answer keys

Concept

The application template

Nearly every application follows the same structure:

total=lim⁡∑(local contribution)=∫(contribution density) d(input).\text{total}=\lim\sum(\text{local contribution})=\int(\text{contribution density})\,d(\text{input}).

The hard part is choosing the contribution correctly. Once the model is right, the calculus is often routine.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.