Problem 11
Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad y=0, \quad x=0, \quad x=1 $$
Problem 11
In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{1}{2}\left(e^{x}+e^{-x}\right), \quad[0,2] $$
Problem 12
Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=\left\\{\begin{array}{ll} \frac{\sin x}{x}, & x>0 \\ 1, & x=0 \end{array}, \quad y=0, \quad x=0, \quad x=\pi\right. $$
Problem 12
In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\ln \left(\frac{e^{x}+1}{e^{x}-1}\right), \quad[\ln 2, \ln 3] $$
Problem 12
A differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. $$ \frac{d y}{d x}=x y, \quad\left(0, \frac{1}{2}\right) $$
Problem 12
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\). $$ y=\frac{1}{2} x^{3}, \quad y=4, \quad x=0 $$
Problem 13
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\). $$ y=\frac{1}{1+x}, \quad y=0, \quad x=0, \quad x=3 $$
Problem 13
Find the function \(y=f(t)\) passing through the point (0,10) with the given first derivative. Use a graphing utility to graph the solution. $$ \frac{d y}{d t}=\frac{1}{2} t $$
Problem 13
In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ x=\frac{1}{3}\left(y^{2}+2\right)^{3 / 2}, \quad 0 \leq y \leq 4 $$
Problem 13
In Exercises \(13-26,\) sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=\frac{1}{2} x^{3}+2, y=x+1, x=0, x=2 $$
