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04:48
04:48
07:36Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.
Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.
b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?
Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).
b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.
Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.
b. When, if ever, does Alicia overtake Boris?
Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.
b. How far does the probe fall in the first 30 s after it is released?
Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.
b. Repeat part (a) using the disk method.
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
b. Use the washer method to write an integral for the volume of the torus.
