← Calculus II

Unit 10: Areas and Volumes

Turning a slice, a shell, or a stretched-out curve into an integral you can actually solve.

6 topics · 27 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Area Between Two Curves

  1. How it worksHow does the top function minus the bottom function on each thin strip build the whole area integral?
  2. Worked exampleFind the area between y = x2 and y = x + 2.
  3. Common mistakeWhy doesn't the area between two curves equal the area under the top curve minus the area under the x-axis?
  4. Going furtherWhat happens to the area-between-curves setup when the curves are given as functions of y instead of x?
Your turn1 of 2
Find the area between y = x2 and y = x + 4.

Area Between Curves That Cross, and Integrating in y

  1. How it worksHow does finding the intersection points first tell you exactly where to break the integral apart?
  2. Worked exampleFind the area between y = sin(x) and y = cos(x) from x = 0 to x = π.
  3. Common mistakeWhy doesn't integrating with respect to x always work, even when a region is easier to describe in y?
  4. Worked exampleFind the area between x = y2 and x = y + 2 by integrating with respect to y instead of x.
Your turn1 of 2
Find the area between y = 4sin(x) and y = 4cos(x) from x = 0 to x = π.

Volumes by Disks and Washers

  1. How it worksHow does subtracting an inner radius from an outer radius before squaring turn a disk into a washer?
  2. Worked exampleFind the volume generated by revolving y = √x from x = 0 to x = 4 around the x-axis.
  3. Worked exampleFind the volume generated by revolving the region between y = x and y = x2 around the x-axis.
  4. Common mistakeWhy doesn't a washer's volume come from squaring the difference of the radii instead of subtracting squares?
  5. Going furtherWhat happens to the disk and washer setup when a region is revolved around a horizontal line other than the x-axis?
Your turn1 of 2
A region under y = √4x is revolved around the x-axis. Write the disk's cross-sectional area A(x), the integrand for the volume integral, as a function of x.

Volumes by Cylindrical Shells

  1. How it worksHow does unrolling one thin cylindrical shell flatten it into a rectangle you can integrate?
  2. Worked exampleFind the volume generated by revolving the region under y = x2 from x = 0 to x = 2 around the y-axis using shells.
  3. How it worksHow does the shell method avoid solving for x in terms of y when a curve is only easy to write as y in terms of x?
  4. Common mistakeWhy doesn't the shell method give a different actual volume than the disk method on the same solid?
  5. Going furtherWhat happens to the shell's radius and height when the axis of revolution is shifted away from the y-axis?
Your turn1 of 2
The region under y = 3x2 from x = 0 is revolved around the y-axis using shells. Write the shell integrand (circumference times height), as a function of x.

Arc Length of a Curve

  1. How it worksHow does chaining together many short straight segments end to end, then shrinking each one, approximate a curve's length?
  2. How it worksHow does the Pythagorean theorem on a tiny piece of a curve turn into the arc length integral of √(1 + (dy/dx)2)?
  3. Worked exampleFind the arc length of y = x3/2 from x = 0 to x = 4.
  4. Common mistakeWhy doesn't the arc length formula just integrate the function itself instead of a square root expression?
  5. Going furtherWhat happens to the arc length integral when the curve is given parametrically instead of as y in terms of x?
Your turn1 of 2
For y = 2x1.5, the arc-length integral is the integral of √(1+(y')2) dx. Write this integrand as a function of x.

Surface Area of a Solid of Revolution

  1. How it worksHow does multiplying an arc length element by the circumference it sweeps out build the surface-area integral?
  2. Common mistakeWhy doesn't the surface area of revolution just equal the curve's arc length times the distance around the axis at one point?
  3. Worked exampleFind the surface area generated by revolving y = √x from x = 1 to x = 4 around the x-axis.
  4. Worked exampleFind the surface area generated by revolving y = x3 from x = 0 to x = 1 around the x-axis.
Your turn1 of 2
For y = 2√x, the surface-area integral is the integral of 2·π·y*√(1+(y')2) dx. Write this integrand as a function of x.