← Math 21a: Multivariable Calculus

Unit 6: Triple Integrals

Volume, mass, and rotational inertia swept through solid regions, in Cartesian, cylindrical, and spherical coordinates.

10 topics · 35 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.

6.1 Triple Integrals over Boxes

  1. How it worksHow does integrating a function three times in a row, once for each variable, add up mass throughout a solid box?
  2. Common mistakeWhy doesn't the order you integrate x, y, and z in matter for a box with constant limits?
  3. Worked exampleFind the triple integral of f(x, y, z) = xyz over the box 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3.
Your turn1 of 2
For the triple integral of f(x, y, z) = 4·x·y·z over the box 0 ≤ z ≤ 3 (x, y held fixed), integrate with respect to z first. Write the result as a function of x and y.

6.2 Triple Integrals over General Solid Regions

  1. How it worksHow do you read a solid's boundary surfaces to set up the innermost, middle, and outermost limits of a triple integral?
  2. Worked exampleFind the volume of the solid tetrahedron bounded by x + y + z = 1 and the three coordinate planes.
  3. Going furtherWhat happens to the limits of a triple integral when you change which variable gets integrated first?
Your turn1 of 2
For the tetrahedron 2·x + 2·y + 3·z ≤ 1 (x, y, z ≥ 0), integrate f(x,y,z)=1 with respect to z from 0 up to its top boundary. Write the result as a function of x and y.

6.3 Triple Integrals in Cylindrical Coordinates

  1. How it worksHow does the extra factor of r appear when a triple integral converts from dz dy dx into dz r dr dtheta?
  2. Common mistakeWhy doesn't converting to cylindrical coordinates skip the extra factor of r, even though only two of the three variables changed?
  3. Worked exampleFind the volume of the cylinder x2 + y2 ≤ 4, 0 ≤ z ≤ 5 using a triple integral in cylindrical coordinates.
Your turn1 of 2
Converting the triple integral of 4·z to cylindrical coordinates, dV becomes r dz dr dtheta. Write the resulting full integrand 4·z·r, as a function of z and r.

6.4 Triple Integrals in Spherical Coordinates

  1. How it worksHow does the factor rho2 sin(phi) appear when a triple integral converts from Cartesian into spherical coordinates?
  2. Worked exampleFind the volume of the solid ball x2 + y2 + z2 ≤ 9 using a triple integral in spherical coordinates.
  3. Common mistakeWhy doesn't the spherical volume element rho2 sin(phi) d rho d phi d theta simplify down to just d rho d phi d theta?
  4. Going furtherWhat happens to the limits of a spherical triple integral when the solid in question is only the top half of the ball?
Your turn1 of 2
Converting the triple integral of 4 to spherical coordinates, dV becomes rho2·sin(phi) d rho d phi d theta. Write the resulting integrand 4·rho2·sin(phi), as a function of rho and phi.

6.5 Mass and Center of Mass in Space

  1. How it worksHow does integrating a density function over a solid region add up all its infinitesimal pieces of mass in three dimensions?
  2. Worked exampleFind the mass of the solid box 0 ≤ x, y, z ≤ 1 with density rho(x, y, z) = x + y + z.
  3. Ties togetherHow does finding the center of mass of a solid in three dimensions extend the same idea used for a flat plate in the plane?
Your turn1 of 2
A solid has density rho(x, y, z) = 4·(x + y + z) over 0 ≤ z ≤ 3 (x, y held fixed). Integrate with respect to z first. Write the result as a function of x and y.

6.6 Moments of Inertia

  1. How it worksHow does the moment of inertia integral weight each piece of mass by the square of its distance from the axis of rotation?
  2. Common mistakeWhy doesn't doubling an object's total mass automatically double its moment of inertia about a given axis?
  3. Worked exampleFind the moment of inertia about the z-axis of a solid cylinder x2 + y2 ≤ 1, 0 ≤ z ≤ 1 with constant density 1.
Your turn1 of 2
Find the moment of inertia about the z-axis of the solid cylinder x2 + y2 ≤ 7, 0 ≤ z ≤ 3, with constant density 4.

6.7 Change of Variables in Multiple Integrals

  1. How it worksHow does a change of variables like u = x + y, v = x - y reshape both the region and the integrand of a double integral?
  2. Common mistakeWhy can't you substitute new variables into a multiple integral without also rescaling the tiny area or volume element?
  3. Worked exampleFind the double integral of (x + y) ex - y over the square with vertices (0,0), (1,1), (2,0), (1,-1), using the substitution u = x + y, v = x - y.
Your turn1 of 2
Under the substitution u = x + y, v = x - y, the integrand 4·(x+y)·ex-y becomes 4·u·ev, and the Jacobian factor is 1/2. Write the transformed integrand including the Jacobian factor, as a function of u and v.

6.8 The Jacobian as a Local Area/Volume Stretch Factor

  1. How it worksHow does the Jacobian determinant measure exactly how much a transformation stretches or shrinks area at each point?
  2. Worked exampleFind the Jacobian of the transformation x = r cos(theta), y = r sin(theta), and confirm it equals r.
  3. Common mistakeWhy doesn't a transformation with Jacobian equal to 1 everywhere have to leave every shape completely unchanged?
  4. Going furtherWhat happens to a region's area under a transformation at a point where the Jacobian determinant equals zero?
Your turn1 of 2
Write the Jacobian of the linear transformation x = a·u + b·v, y = c·u + d·v, as a function of a, b, c, and d.

6.9 Choosing Cylindrical vs Spherical Coordinates

  1. How it worksHow do you decide between cylindrical and spherical coordinates based on whether a solid's boundary is built from planes and cylinders or from cones and spheres?
  2. Worked exampleBuild the triple integral for the solid bounded above by a sphere and below by a cone, after deciding between cylindrical and spherical coordinates.
  3. Common mistakeWhy doesn't cylindrical coordinates work just as smoothly for a solid ball as spherical coordinates do?
  4. Ties togetherHow does this choice between coordinate systems mirror the same kind of decision made earlier when choosing between direct computation and a coordinate-based shortcut?
Your turn1 of 2
A solid is bounded above by the sphere rho = 4 and below by the cone phi = 0.95 (in radians). Find its volume using spherical coordinates.

6.10 A Complete Worked Setup for a Triple Integral

  1. How it worksHow do you read a solid region's description to decide the order of integration and the limits for each of the three variables in turn?
  2. Worked exampleBuild and evaluate the triple integral for the mass of the solid bounded by z = 0 and z = 4 - x2 - y2, using density rho(x, y, z) = 1.
  3. Worked exampleBuild the same triple integral from the previous step using cylindrical coordinates instead, and confirm the two setups agree.
  4. Common mistakeWhy doesn't a correctly set up triple integral have to look the same in Cartesian and in cylindrical coordinates, even though both give the same final answer?
  5. Going furtherWhat happens to the setup when the solid's top surface z = 4 - x2 - y2 is replaced by a flat plane instead?
Your turn1 of 2
Find the mass of the solid bounded by z = 0 and z = 14 - 4·(x2+y2), using density rho(x, y, z) = 4.