MA 227 Standard C08

Triple Integrals

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)