Chapter 7: Problem 48
Find the area of the propeller-shaped region enclosed between the graphs of $$y=\sin x$$ and $$y=x^{3}$$ .
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Multiple Choice Which of the following expressions should be used to find the length of the curve \(y=x^{2 / 3}\) from \(x=-1\) to \(x=1 ?\) \((\mathbf{A}) 2 \int_{0}^{1} \sqrt{1+\frac{9}{4} y} d y \quad\) (B) \(\int_{-1}^{1} \sqrt{1+\frac{9}{4} y} d y\) (C) \(\int_{0}^{1} \sqrt{1+y^{3}} d y \quad\) (D) \(\int_{0}^{1} \sqrt{1+y^{6}} d y\) \((\mathbf{E}) \int_{0}^{1} \sqrt{1+y^{9 / 4}} d y\)
Writing to Learn You are in charge of the evacuation and repair of the storage tank shown here. The tank is a hemisphere of radius 10 \(\mathrm{ft}\) and is full of benzene weighing 56 \(\mathrm{lb} / \mathrm{ft}^{3}\) . A firm you contacted says it can empty the tank for 1\(/ 2\) cent per foot-pound of work. Find the work required to empty the tank by pumping the benzene to an outlet 2 ft above the tank. If you have budgeted \(\$ 5000\) for the job, can you afford to hire the firm?
The region bounded below by the parabola $$y=x^{2}$$ and above $$b$$ . the line $$y=4$$ is to be partitioned into two subsections of equal area by cutting across it with the horizontal line $$y=c$$. (a) Sketch the region and draw a line $$y=c$$ across it that looks about right. In terms of $$c,$$ what are the coordinates of the points where the line and parabola intersect? Add them to your figure. (b) Find $$c$$ by integrating with respect to y. (This puts $$c$$ in the limits of integration. (c) Find $$c$$ by integrating with respect to $$x$$ . (This puts $$c$$ into the integrand as well.)
Putting a Satellite into Orbit The strength of Earth's gravitational field varies with the distance \(r\) from Earth's center, and the magnitude of the gravitational force experienced by a satellite of mass \(m\) during and after launch is $$F(r)=\frac{m M G}{r^{2}}$$ Here, \(M=5.975 \times 10^{24} \mathrm{kg}\) is Earth's mass, \(G=6.6726 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} \mathrm{kg}^{-2}\) is the universal gravitational constant, and \(r\) is measured in meters. The work it takes to lift a 1000 -kg satellite from Earth's surface to a circular orbit \(35,780 \mathrm{km}\) above Earth's center is therefore given by the integral Work $$=\int_{6,370,000}^{35.780,000} \frac{1000 M G}{r^{2}} d r$$ joules. The lower limit of integration is Earth's radius in meters at the launch site. Evaluate the integral. (This calculation does not take into account energy spent lifting the launch vehicle or energy spent bringing the satellite to orbit velocity.)
In Exercises 39-42, find the volume of the solid analytically. The base of the solid is the disk \(x^{2}+y^{2} \leq 1 .\) The cross sections by planes perpendicular to the \(y\) -axis between \(y=-1\) and \(y=1\) are isosceles right triangles with one leg in the disk.
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