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Question

# Determine which value best approximates the volume of the solid between the $x y$ plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \$f(x, y)=\sqrt{x^2+y^2}$$R$ : circle bounded by $x^2+y^2=9$ (a) $50$ (b) $500$ (c ) $-500$ (d) $5$ (e) $5000$

Solution

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First recognize the given function as a cone. As soon as the value of function exceeds $9$ we are out of the boundaries. The volume of the shape is approximately equal to area of the base (the circle we are given) times the height of the shape. However, the volume we are estimating is $\textbf{below}$ the graph of the given function, consequently, you should think about it as part of the cylinder that remains once you cut out cone. Volume is positive value, so once again we ignore the negative answer. Since the area of the base is already approximately equal to $28.3$ the answer $5$ is out of the consideration. Lastly, do note that the function $f(x,y)=\sqrt{x^2+y^2}$ increases moderately slow (slower than the linear function) hence the only possible approximation should be 50.

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