삼중적분

Let $F,G,\,\text{and}\,H$ be continuous functions on $[a,b],[c,d],$ and $[e,f],$ respectively, where $a,b,c,d,e,\,\text{and}\,f$ are real numbers such that $a<b,c<d,\,\text{and}\,e<f.$ Show that

$$\int_{a}^{b}\,\int_{c}^{d}\,\int_{e}^{f}F(x)G(y)H(z)dz\,dy\,dx=\left(\int_{a}^{b}F(x)dx\right)\left(\int_{c}^{d}G(y)dy\right)\left(\int_{e}^{f}H(z)dz\right).$$

Let $F,G,\,\text{and}\,H$ be differential functions on $[a,b],[c,d],$ and $[e,f],$ respectively, where $a,b,c,d,e,\,\text{and}\,f$ are real numbers such that $a<b,c<d,\,\text{and}\,e<f.$ Show that

$$\int_{a}^{b}\,\int_{c}^{d}\,\int_{e}^{f}{F}^{'}(x){G}^{'}(y){H}^{'}(z)dz\,dy\,dx=[F(b)-F(a)]\,[G(d)-G(c)]\,[H(f)-H(e)].$$

Consider the pyramid with the base in the $xy$-plane of $[-2,2]\,\times \,[-2,2]$ and the vertex at the point $(0,0,8).$

a. Show that the equations of the planes of the lateral faces of the pyramid are $4y+z=8,$ $4y-z=-8,$ $4x+z=8,$ and $-4x+z=8.$ b. Find the volume of the pyramid.

Consider the pyramid with the base in the $xy$-plane of $[-3,3]\,\times \,[-3,3]$ and the vertex at the point $(0,0,9).$

a. Show that the equations of the planes of the side faces of the pyramid are $$3x+z=9,-3x+z=9,-3y+z=9,\text{ and }3y+z=9$$ b. Find the volume of the pyramid.

Show that the volume of a right square pyramid of height $h$ and side length $a$ is $v=\frac{h{a}^{2}}{3}$ by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length $a$ is $\frac{3{a}^{3}\sqrt{3}}{2}$ by using triple integrals.

Show that the volume of a regular right hexagonal pyramid of edge length $a$ is $\frac{{a}^{3}\sqrt{3}}{2}$ by using triple integrals.