Chapter 12: Problem 4
In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u v-2 u, y=u v\)
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In Exercises \(59-62,\) use a computer algebra system to approximate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{1+\cos \theta} 6 r^{2} \cos \theta d r d \theta $$
Find the mass and center of mass of the lamina for each density. \(R:\) triangle with vertices \((0,0),(0, a),(a, 0)\) (a) \(\rho=k\) (b) \(\rho=x^{2}+y^{2}\)
Find \(k\) such that the function \(f(x, y)=\left\\{\begin{array}{ll}k e^{-\left(x^{2}+y^{2}\right)}, & x \geq 0, y \geq 0 \\ 0, & \text { elsewhere }\end{array}\right.\) is a probability density function.
Approximation \(\quad\) In Exercises 41 and 42, 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.) \(f(x, y)=15-2 y ; R:\) semicircle: \(x^{2}+y^{2}=16, y \geq 0\) (a) 100 (b) 200 (c) 300 (d) -200 (e) 800
In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{0}^{4} \int_{0}^{y} f(x, y) d x d y $$
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