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Activity 12.6.4.

There are several other ways we could have set up the integral to give the mass of the tetrahedron in Example 12.6.6.

(a)

How many different orders of integration could be used for iterated integrals that are equal to the integral in Equation (12.6.2)?

(b)

Set up an iterated integral, integrating first with respect to \(z\text{,}\) then \(x\text{,}\) then \(y\) that is equivalent to the integral in Equation (12.6.2). Before you write down the integral, think about Figure 12.6.7, and draw an appropriate two-dimensional image of an important projection.

(c)

Set up an iterated integral, integrating first with respect to \(y\text{,}\) then \(z\text{,}\) then \(x\) that is equivalent to the integral in Equation (12.6.2). As in (b), think carefully about the geometry first and draw a plot of the appropriate projection.

(d)

Set up an iterated integral, integrating first with respect to \(x\text{,}\) then \(y\text{,}\) then \(z\) that is equivalent to the integral in Equation (12.6.2).