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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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In polar coordinates an equation of an ellipse with eccentricity \(0 < e < 1\) and semimajor axis \(a\) is \(r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}\) a. Write the integral that gives the area of the ellipse. b. Show that the area of an ellipse is \(\pi a b,\) where \(b^{2}=a^{2}\left(1-e^{2}\right)\)
Intersecting spheres One sphere is centered at the origin and has a radius of \(R\). Another sphere is centered at \((0,0, r)\) and has a radius of \(r,\) where \(r>R / 2 .\) What is the volume of the region common to the two spheres?
Find the coordinates of the center of mass of the following solids with variable density. The interior of the prism formed by \(z=x, x=1, y=4,\) and the coordinate planes with \(\rho(x, y, z)=2+y\)
Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of the cap of a sphere of radius \(R\) with height \(h\)
The occurrence of random events (such as phone calls or e-mail messages) is often idealized using an exponential distribution. If \(\lambda\) is the average rate of occurrence of such an event, assumed to be constant over time, then the average time between occurrences is \(\lambda^{-1}\) (for example, if phone calls arrive at a rate of \(\lambda=2 /\) min, then the mean time between phone calls is \(\lambda^{-1}=\frac{1}{2} \mathrm{min}\) ). The exponential distribution is given by \(f(t)=\lambda e^{-\lambda t},\) for \(0 \leq t<\infty\) a. Suppose you work at a customer service desk and phone calls arrive at an average rate of \(\lambda_{1}=0.8 /\) min (meaning the average time between phone calls is \(1 / 0.8=1.25 \mathrm{min}\) ). The probability that a phone call arrives during the interval \([0, T]\) is \(p(T)=\int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} d t .\) Find the probability that a phone call arrives during the first 45 s \((0.75\) min) that you work at the desk. b. Now suppose that walk-in customers also arrive at your desk at an average rate of \(\lambda_{2}=0.1 /\) min. The probability that a phone $$p(T)=\int_{0}^{T} \int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} \lambda_{2} e^{-\lambda_{2} x} d t d s$$ Find the probability that a phone call and a customer arrive during the first 45 s that you work at the desk. c. E-mail messages also arrive at your desk at an average rate of \(\lambda_{3}=0.05 /\) min. The probability that a phone call and a customer and an e-mail message arrive during the interval \([0, T]\) is $$p(T)=\int_{0}^{T} \int_{0}^{T} \int_{0}^{T} \lambda_{1} e^{-\lambda_{1} t} \lambda_{2} e^{-\lambda_{2} s} \lambda_{3} e^{-\lambda_{3} u} d t d s d u$$ Find the probability that a phone call and a customer and an e-mail message arrive during the first 45 s that you work at the desk.
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