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Activity 12.3.5.

Consider the double integral \(\displaystyle{\iint_D (4-x-2y) \, dA}\text{,}\) where \(D\) is the triangular region with vertices (0,0), (4,0), and (0,2).

(a)

Draw and label a plot of \(D\) with relevent cross sections for a vertically simple description. Give the inequalities that show \(D\) as vertically simple.

(b)

Draw and label a plot of \(D\) with relevent cross sections for a horizontally simple description. Give the inequalities that show \(D\) as horizontally simple.

(c)

Write the double integral as an iterated integral of the form \(\iint_D (4-x-2y) \, dy \, dx\) and include bounds on both of your iterated integrals.

(d)

Write the double integral as an iterated integral of the form \(\iint_D (4-x-2y) \, dx \, dy\) and include bounds on both of your iterated integrals.

(e)

Evaluate the two iterated integrals from the previous two parts, and verify that they produce the same value. Give at least one interpretation of the meaning of your result.