# Express the triple integral over E of f(x, y, z) dV as an iterated integral in six different...

## Question:

Express {eq}\; \iiint_{E} f(x, y, z) \, \mathrm{d}V \; {/eq} as an iterated integral in six different ways, where {eq}E {/eq} is the solid bounded by {eq}\; z = 0, \; x = 0, \; z = y - x {/eq} and {eq}\; y = 3 {/eq}.

## Expressing Triple Integrals as Iterated Integrals:

Consider the integral of the function {eq}f(x,y,z) {/eq} over the solid {eq}E {/eq}, written as the triple integrals:

$$\iiint_E f(x,y,z) \,dV$$

The triple integrals shown above can be expressed as iterated integrals in six different ways depending on the order of integration which is reflected on the arrangement of the elements of the differential volume {eq}dV = dx \,dy \,dz {/eq}.

Suppose the solid {eq}E {/eq} is bounded above and below by the surfaces {eq}z = h_2(x,y) {/eq} and {eq}z = h_1(x,y) {/eq}, respectively, over the region {eq}D {/eq} on the {eq}xy {/eq}-plane.

Additionally, suppose {eq}D {/eq} is bounded above and below by the curves {eq}y = g_2(x) {/eq} and {eq}y = g_1(x) {/eq}, respectively, over the interval {eq}a \le x \le b {/eq}.

Then, one of the six iterated integrals of the given triple integrals is:

$$\iiint_E f(x,y,z) \,dV = \iint_D \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \,dz \,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \,dz \,dy \,dx$$