Triple Integrals

Trivium

Given a solid EE in R3\mathbf{R}^3 with a boundary that can be described analytically either in rectangular, cylindrical, or spherical coordinates, and a function f ⁣:ER,f\colon E \to \mathbf{R}, compute the value of EfdV. \iiint\limits_E f \,\mathrm{d}V\,.

Given an iterated integral EfdV\iiint_E f \,\mathrm{d}V of a function f ⁣:ERf\colon E \to \mathbf{R} on a region ER3E \subset \mathbf{R}^3 and a C1,C^1, one-to-one transformation TT of R3,\mathbf{R}^3, compute the Jacobian of the transformation JT\left|\operatorname{J}_T\right| and write down the equivalent iterated integral over the pre-transformed domain, T1(E)(fT)JTdV. \iiint\limits_{T^{-1}(E)} \big(f \circ T\big) \,\bigl|\operatorname{J}_{T}\bigr| \,\mathrm{d}V\,.

Exercises

  1. Evaluate the triple integral Sxyz2dV\iiint_S xyz^2 \,\mathrm{d}V where SS is the rectangular prism S=[0,1]×[1,2]×[0,3].S = [0,1]\times[-1,2]\times[0,3]\,.
  2. Evaluate SzdV\iiint_S z \,\mathrm{d}V where SS is the solid in the first octant bounded by the surface z=12xyz = 12xy and the planes y=xy=x and x=1.x=1.
  3. Evaluate Sx2+z2dV\iiint_S \sqrt{x^2+z^2} \,\mathrm{d}V where SS is the solid bounded by the paraboloid y=x2+z2y = x^2+z^2 and the plane y=4.y=4.
  4. Sketch the solid over which the iterated integral 010x20yf(x,y,z)dzdydx\int_0^{1}\int_0^{x^2}\int_0^{y} f(x,y,z) \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x is concerned, and then rewrite the iterated integral with the order of the variables permuted: first with respect to dxdzdy,\mathrm{d}x\,\mathrm{d}z\,\mathrm{d}y, then with respect to dydxdz.\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}z.
  5. Calculate the volume of the tetrahedron bound by the planes x+2y+z=2x+2y+z=2 and x=2yx=2y and x=0x=0 and z=0.z=0.
  6. Calculate the coordinates of the center of mass of the solid with constant density that is bounded by the parabolic cylinder x=y2x=y^2 and the planes x=zx=z and z=0z=0 and x=1.x=1.
  7. Evaluate Sx2dV\iiint_S x^2 \,\mathrm{d}V for the solid SS that lies under the paraboloid z=4x2y2z=4-x^2-y^2 and above the xyxy-plane.
  8. A solid SS lies within the cylinder x2+y2=1x^2+y^2=1 on the positive yy side of the xzxz-plane, below the plane z=4,z=4, and above the paraboloid z=1x2y2.z=1-x^2-y^2. The density at any point in the solid is three times its distance to the zz-axis. What is the mass of S?S?
  9. Evaluate this iterated integral:
    224x24x2x2+y22(x2+y2)dzdydx\displaystyle \int\limits_{-2}^{2} \int\limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int\limits_{\sqrt{x^2+y^2}}^{2} \Big(x^2 + y^2\Big) \,\mathrm{d}z \,\mathrm{d}y \,\mathrm{d}x
  10. For the unit ball S={(x,y,z)x2+y2+z21}S = \big\{(x,y,z) \mid x^2+y^2+z^2 \leq 1 \big\} evaluate
    Se(x2+y2+z2)3/2dV\displaystyle \iiint_S \mathrm{e}^{(x^2+y^2+z^2)^{3/2}} \,\mathrm{d}V
  11. Calculate the volume of the solid SS that lies above the cone z=x2+y2z = \sqrt{x^2+y^2} but below the sphere x2+y2+z2=z.x^2+y^2+z^2=z.
  12. Consider the following change of coordinates (ρ,θ,ϕ)(x,y,z)(\rho, \theta, \phi) \to (x,y,z) and calculate the corresponding Jacobian.
    x=ρsin(ϕ)cos(θ)\displaystyle x=\rho\sin(\phi)\cos(\theta)
    y=ρsin(ϕ)sin(θ)\displaystyle y=\rho\sin(\phi)\sin(\theta)
    z=ρcos(ϕ)\displaystyle z=\rho\cos(\phi)

Problems & Challenges

  1. Consider the tetrahedron in R3\mathbf{R}^3 with vertices (0,0,0)(0,0,0) and (0,0,1)(0,0,1) and (0,3,1)\big(0,\sqrt{3},-1\big) and (3,3,2)\big(3,\sqrt{3},-2\big).
    1. Write down three integrals — one each based on rectangular, cylindrical, and spherical coordinates — that express the volume of the tetrahedron. Then evaluate one of them to calculate the volume.
    2. What are the coordinates of the centroid of this tetrahedron?
    3. What is the surface area of this tetrahedron?
    4. There exists a change of variables, a transformation of space, that maps the vertices of this tetrahedron to the points (0,0,0)(0,0,0) and (0,0,1)(0,0,1) and (0,1,0)(0,1,0) and (1,0,0)(1,0,0). What is this transformation, and what is its Jacobian? Use the transformation and Jacobian to compute the volume of the original tetrahedron.
  2. James Stewart Show that 01010111xyzdxdydz=n=11n3 \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xyz} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \sum_{n=1}^{\infty} \frac{1}{n^3} So far, no one has been able to express the value of this integral/sum exactly in terms of already-known constants.