Unit 4 of 7

Geometric construction

  • 5 chapters
  • Chapters 15 to 19

Each integral here is built from geometric pieces. Strips give the area between curves, washers and shells give volumes of revolution, and short segments and bands give arc length and surface area.

Each chapter starts from the picture, chooses a small piece whose size is known and adds the pieces with an integral. The area between a line and a parabola is the integral of top minus bottom between their intersections. Rotating a region about an axis gives washers when the strips are perpendicular to the axis and cylindrical shells when they are parallel to it. The Pythagorean theorem gives the length of a short piece of a curve in terms of the derivative, and rotating that piece gives a band whose area is the circumference times the slant length. Simpler shapes bound each answer.

Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.

Area between curves and volumes by washers and shells

  1. Build the integral for a region between curves

    How to build the area between two curves from thin strips, with intersection points as limits, top minus bottom as the strip height, vertical or horizontal strips, and the absolute difference when curves cross.

    Chapter 15

  2. Build a solid from washers

    How the washer method finds the volume of a solid of revolution by slicing it perpendicular to the axis, where each cross-sectional area is pi times the outer radius squared minus the inner radius squared.

    Chapter 16

  3. Build a solid from cylindrical shells

    How the shell method finds the volume of a solid of revolution from strips parallel to the axis, each sweeping a thin cylindrical shell of about circumference times height times thickness, with rotation about the y-axis.

    Chapter 17

Arc length and surface area of revolution

  1. Measure the length of a curve

    How the Pythagorean length of short chords leads to the arc length element and the arc length integral, why the curve must be continuously differentiable, and how to find an exact arc length.

    Chapter 18

  2. Measure a curved surface in three dimensions

    How the surface area of revolution is built from thin bands close to conical frustums, why the surface area element is the circumference times the slant length, and how to find the area of a lateral surface.

    Chapter 19