This problem is based on Example #1 in §12.3 of Stewart (pg 846).

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The goal is to evaluate $\displaystyle \int\int_{{D}}{\left({7}{x}+{4}{y}\right)}{d}{A}$, where $\displaystyle {D}$ is the region bounded by the parabolas $\displaystyle {y}={3}{x}^{{2}}$ and $\displaystyle {y}={8}+{x}^{{2}}$.

Against which variable will you integrate first, i.e., what is the dummy variable for the inner-integral?


What are the bounds of integration for the inner integral?
lower bound = Preview Question 6 Part 2 of 7  
upper bound = Preview Question 6 Part 3 of 7  

What are the bounds of integration for the outer integral?
lower bound =
upper bound =

Work the first (inner) integral. This should result in an integral of the form $\displaystyle {\int_{{a}}^{{b}}}{g{{\left({z}\right)}}}{\left.{d}{z}\right.}$, where $\displaystyle {z}$ is the appropriate variable ($\displaystyle {x}$ or $\displaystyle {y}$). What is the function in this integrand?
the integrand is Preview Question 6 Part 6 of 7  
Hint: Look at the penultimate line in the calculation for this example.

Finally, $\displaystyle \int\int_{{D}}{\left({7}{x}+{4}{y}\right)}{d}{A}=$ Preview Question 6 Part 7 of 7