Chapter 13: Problem 53
Use double integrals to calculate the volume of the following regions. The tetrahedron bounded by the coordinate planes \((x=0, y=0, z=0)\) and the plane \(z=8-2 x-4 y\)
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Use a change of variables to evaluate the following integrals. $$\begin{aligned} &\iiint_{D} x y d V ; D \text { is bounded by the planes } y-x=0\\\ &y-x=2, z-y=0, z-y=1, z=0, \text { and } z=3 \end{aligned}$$
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A region is enclosed by an isosceles triangle with two sides of length \(s\) and a base of length \(b\). How far from the base is the center of mass?
A disk of radius \(r\) is removed from a larger disk of radius \(R\) to form an earring (see figure). Assume the earring is a thin plate of uniform density. a. Find the center of mass of the earring in terms of \(r\) and \(R\) (Hint: Place the origin of a coordinate system either at the center of the large disk or at \(Q\); either way, the earring is symmetric about the \(x\) -axis.) b. Show that the ratio \(R / r\) such that the center of mass lies at the point \(P\) (on the edge of the inner disk) is the golden mean \((1+\sqrt{5}) / 2 \approx 1.618\)
Let \(f\) be a continuous function on \([0,1] .\) Prove that $$\int_{0}^{1} \int_{x}^{1} \int_{x}^{y} f(x) f(y) f(z) d z d y d x=\frac{1}{6}\left(\int_{0}^{1} f(x) d x\right)^{3}$$
Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Find the area of \(R\)
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