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MA 227 Standard C08

Triple Integrals
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At the end of the course, each student should be able to…

  • C08: TripleInt. Compute and apply triple integrals.

C08. Triple Integrals

  • The triple integral may be used to find the mass of an object with variable density.
    • Let \(\iiint_D f(x,y,z)\,dV\) be the mass \(M\) of the solid \(D\) with density given by \(f(x,y,z)\) at each point in \(D\).
    • Let \(R\) be the shadow of \(D\) in the \(xy\) plane, and let \(z=h_1(x,y)\) and \(z=h_2(x,y)\) describe the top and bottom surfaces of \(D\). The contribution of mass \(\frac{dM}{dA}\) for each point in \(R\) is given by \(\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z)\,dz\).
    • Thus the triple integral may be evaluated by \(\iiint_D f(x,y,z)\,dV=\iint_R\frac{dM}{dA}\,dA= \iint_R[\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z)\,dz]\,dA\).
    • This expands to the iterated integral \(\int_a^b\int_{g_1(x)}^{g_2(x)}\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) \,dz\,dy\,dx\) where \(g_1(x),g_2(x)\) are the top/bottom curves of the shadow of \(D\) on the \(xy\) plane, and \(h_1(x,y),h_2(x,y)\) are the top/bottom surfaces bounding \(D\) in \(xyz\) space.
  • The properties of double integrals also hold for triple integrals.
    • \(\iiint_D cf\,dV=c\iiint_D f\,dV\).
    • \(\iiint_D (f\pm g)\,dV= \iiint_D f\,dV \pm \iiint_D g\,dV\).
    • \(f\leq g\) implies \(\iiint_D f\,dV\leq\iiint_D g\,dV\).
    • \(\iiint_{D+E} f\,dV = \iiint_D f\,dV + \iiint_E f\,dV\).
  • Triple integrals may be applied to find volumes and average values.
    • The volume of \(D\) is given by \(\iiint_D 1\,dV\).
    • The average value of \(f\) over \(D\) is given by \(\frac{1}{Volume(D)}\iint_D f\,dV\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 14.5 (exercises 1-20, 23-40)