[12.7] - Triple integrals / Volume

  1. Find the volume of the figures shown below, using both a triple integral, and geometry.


  2. Sketch the solid whose volume is given by $$\int_0^4\int_0^{\frac{4-x}{2}} \int_0^{\frac{12-3x-6y}{4}} dz\,dy\,dx$$

    Then rewrite the integral in the order $\int \int \int dy\,dx\,dz$ with the appropriate limits of integration.

  3. Find the volume of the tetrahedron bounded by the coordinate plane $z=4-4x-2y$.

  4. Find the volume of the tetrahedron shown below.

  1. First figure, by geometry $V=\frac12*(2\times1\times1)=1$;
    Setting up the integral... $$\int_{x=0}^2\int_{y=0}^1\int_{z=0}^{1-y} dz\,dy\,dx=\int_{x=0}^2\int_{y=0}^1(1-y) \,dy\,dx=...$$
  2. By geometry $V=4\times2\times2+4\times2\times\frac124=32$
  3. $$\begineq V&=&\iiint dV=\iiint dx\,dy\,dz\\ &=& \int_{x=0}^1\int_0^{y=2-2x}\int_0^{z=4-4x-2y}dz\,dy\,dx\\ &=&\frac43 \endeq $$