Chapter 12: Problem 31
In Exercises \(31-36,\) use an iterated integral to find the area of the region bounded by the graphs of the equations. $$ \sqrt{x}+\sqrt{y}=2, \quad x=0, \quad y=0 $$
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Prove the following Theorem of Pappus: Let \(R\) be a region in a plane and let \(L\) be a line in the same plane such that \(L\) does not intersect the interior of \(R .\) If \(r\) is the distance between the centroid of \(R\) and the line, then the volume \(V\) of the solid of revolution formed by revolving \(R\) about the line is given by \(V=2 \pi r A,\) where \(A\) is the area of \(R\)
In Exercises \(51-54,\) evaluate the iterated integral. (Note that it is necessary to switch the order of integration.) $$ \int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x $$
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{\pi / 4} \int_{0}^{\cos \theta} 3 r^{2} \sin \theta d r d \theta $$
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.) \(f(x, y)=x y+2 ; R:\) quarter circle: \(x^{2}+y^{2}=9, x \geq 0, y \geq 0\) (a) 25 (b) 8 (c) 100 (d) 50 (e) -30
Approximation \(\quad\) In Exercises 39 and \(40,\) use a computer algebra system to approximate the iterated integral. $$ \int_{\pi / 4}^{\pi / 2} \int_{0}^{5} r \sqrt{1+r^{3}} \sin \sqrt{\theta} d r d \theta $$
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