Chapter 12: Problem 48
Determine the value of \(b\) such that the volume of the ellipsoid \(x^{2}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{9}=1\) is \(16 \pi\)
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In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} d y d x $$
Approximation \(\quad\) In Exercises 39 and \(40,\) use a computer algebra system to approximate the iterated integral. $$ \int_{\pi / 4}^{\pi / 2} \int_{0}^{5} r \sqrt{1+r^{3}} \sin \sqrt{\theta} d r d \theta $$
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{\pi} \int_{0}^{\sin x}(1+\cos x) d y d x $$
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.
In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{0}^{2} d y d x $$
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