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Activity 12.3.6 .
Consider the iterated integral
\(\int_{x=0}^{x=1} \int_{y=x}^{y=\sqrt{x}} (4x+10y) \, dy \, dx\text{.}\)
(a)
Sketch the region of integration, \(D\text{,}\) for which
\begin{equation*}
\iint_D (4x + 10y) \, dA = \int_{x=0}^{x=1} \int_{y=x}^{y=\sqrt{x}} (4x+10y) \, dy \, dx.
\end{equation*}
(b)
Determine the equivalent iterated integral that results from integrating in the opposite order (\(dx \, dy\text{,}\) instead of \(dy \, dx\) ). That is, determine the limits of integration for which
\begin{equation*}
\iint_D (4x + 10y) \, dA = \int_{y=?}^{y=?} \int_{x=?}^{x=?} (4x+10y) \, dx \, dy.
\end{equation*}
(c) Evaluate which ever one of the two iterated integrals above you think will be easiest to compute. Explain what the value you obtained tells you.
(d) Set up and evaluate a single definite integral to determine the exact area of
\(D\text{,}\) \(A(D)\text{.}\)
(e) Determine the exact average value of
\(f(x,y) = 4x + 10y\) over
\(D\text{.}\)
