Problem 22
Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=x^{2}, \quad y=4 x-x^{2}, \text { about the line } x=2 $$
Problem 22
Lifting \(a\) Chain In Exercises 19-22, consider a 15-foot chain that weighs 3 pounds per foot hanging from a winch 15 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up the entire chain with a 500 -pound load attached to it.
Problem 23
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x^{2}+1, \quad y=-x^{2}+2 x+5, \quad x=0, \quad x=3 $$
Problem 23
Approximation In Exercises 23 and \(24,\) determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.) \(\int_{0}^{2} \sqrt{1+\left[\frac{d}{d x}\left(\frac{5}{x^{2}+1}\right)\right]^{2}} d x\) (a) 25 (b) 5 (c) 2 (d) -4 (e)
Problem 23
Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=4 x-x^{2}, \quad y=0, \text { about the line } x=5 $$
Problem 23
Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x y $$
Problem 23
Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(y)=y^{2}+1, g(y)=0, \quad y=-1, \quad y=2 $$
Problem 23
Boyle's Law A quantity of gas with an initial volume of 2 cubic feet and a pressure of 1000 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas using the integral \(W=\int_{V_{0}}^{v_{1}} p d V\) Assume that the pressure is inversely proportional to the volume.
Problem 24
Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(y)=\frac{y}{\sqrt{16-y^{2}}}, \quad g(y)=0, \quad y=3 $$
Problem 24
Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=\sqrt{x}, \quad y=0, \quad x=4, \text { about the line } x=6 $$
