Chapter 13: Problem 48
Identify the quadric surface. $$ z^{2}-x^{2}-\frac{y^{2}}{4}=1 $$
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Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{2 y}^{2} d x d y $$
Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{1}^{2} \int_{2}^{4} d x d y $$
Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x $$
Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x y, z=0, y=0, y=4, x=0, x=1 $$
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. $$ (1,4),(2,6),(3,8),(4,11),(5,13),(6,15) $$
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