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Chapter 13: Problem 27
Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\ln (4-x-y) $$
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Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (0.5,2),(0.75,1.75),(1,3),(1.5,3.2),(2,3.7),(2.6,4) $$
Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (1,10.3),(2,14.2),(3,18.9),(4,23.7),(5,29.1),(6,35) $$
Evaluate the partial integral. $$ \int_{1}^{2 y} \frac{y}{x} d x $$
Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=9-x^{2}, y=0 $$
Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int \frac{y}{1+x^{2}} d A\\\ &R: \text { region bounded by } y=0, y=\sqrt{x}, x=4 \end{aligned} $$
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