Multiple integrals · Definition

Triple integral

multivariable calculusmultiple integrals

Before you start you need to know

It accumulates a quantity spread through space, such as mass with variable density or averages of a scalar field over a solid.

Definition

The triple integral of a function \(f\) over a solid \(E\) of space is the limit of the sums of the values of \(f\) times the volume \(\Delta V\) of ever finer cells covering \(E\).

\[\iiint_E f\,dV=\lim_{\max\Delta V_i\to0}\sum_i f(x_i^*,y_i^*,z_i^*)\,\Delta V_i\]

With \(f=1\) the triple integral is the volume of \(E\), and if \(\rho\) is a density, \(\iiint_E\rho\,dV\) is the mass. The average value of \(f\) over \(E\) is \(\tfrac1{V(E)}\iiint_E f\,dV\). As in two variables, the integral is linear and additive with respect to the solid, and exists if \(f\) is continuous and \(E\) has a piecewise smooth boundary.

It is computed with iterated integrals. On a box \([a,b]\times[c,d]\times[p,q]\), Fubini's theorem allows integrating in any of the \(3!=6\) orders with the same result. If \(E\) is the solid \(\{(x,y)\in D,\ u_1(x,y)\le z\le u_2(x,y)\}\), then \(\iiint_E f\,dV=\iint_D\Big[\int_{u_1}^{u_2}f\,dz\Big]dA\): the inner integral runs along a vertical segment that enters at \(u_1\) and leaves at \(u_2\), and the result is integrated over the projection \(D\).

Guiding example: the tetrahedron bounded by the coordinate planes and \(x+y+z=1\) has volume \(\int_0^1\!\int_0^{1-x}\!\int_0^{1-x-y}dz\,dy\,dx=\tfrac16\). The practical method is to sketch the solid and its shadow \(D\) in the \(xy\)-plane, write the limits of \(z\) first using the lower and upper surfaces, and then describe \(D\) as a general region.

Typical mistakes: putting already-integrated variables in the outer limits, mixing up which surface is on top, and forgetting that the order of the differentials must match the order of the limits, from inside out. When the solid has circular or spherical symmetry, it is better to switch to cylindrical or spherical coordinates.

Think about it

What does \(\iiint_E 1\,dV\) represent?

The volume of the solid \(E\).

Write the limits of \(z\) for the solid between the plane \(z=0\) and the paraboloid \(z=4-x^2-y^2\).

\(0\le z\le 4-x^2-y^2\), integrating \(x,y\) over the disc \(x^2+y^2\le4\).

Exercises 10

Compute the following integrals

  1. \(\displaystyle \int_0^1\!\int_0^1\!\int_0^1 xyz\,dz\,dy\,dx\)
  2. \(\displaystyle \int_0^1\!\int_0^2\!\int_0^3 (x+y+z)\,dz\,dy\,dx\)
  3. \(\displaystyle \int_0^1\!\int_0^x\!\int_0^y x\,dz\,dy\,dx\)

Problems

  1. Compute \(\iiint_B 1\,dV\) over the box \(B=[0,1]\times[0,2]\times[0,3]\).
  2. The tetrahedron \(E\) is bounded by the coordinate planes and \(x+y+z=1\). Its volume \(\int_0^1\!\int_0^{1-x}\!\int_0^{1-x-y}dz\,dy\,dx\) is:
  3. With the same tetrahedron \(E\), compute \(\iiint_E z\,dV\).
  4. A cube \([0,2]^3\) has density \(\rho(x,y,z)=x\). Its mass \(\iiint\rho\,dV\) is:
  5. Find the volume of the solid in the first octant under the plane \(x+2y+3z=6\).
  6. The average value \(\tfrac1{V(E)}\iiint_E f\,dV\) of \(f=x+y+z\) over the unit cube \(E=[0,1]^3\) is:
  7. For a continuous function on a box, how many iterated orders of integration are there and how are their results related?

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §15.6
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §15.5
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §5.5