Problem 10
Write and solve the differential equation that models the verbal statement. The rate of change of \(P\) with respect to \(t\) is proportional to \(10-t\).
Problem 10
In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\ln (\cos x), \quad\left[0, \frac{\pi}{3}\right] $$
Problem 10
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. $$ y=2 x, \quad y=4, \quad x=0 $$
Problem 10
Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. Seven and one-half foot-pounds of work is required to compress a spring 2 inches from its natural length. Find the work required to compress the spring an additional one-half inch.
Problem 10
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=6-2 x-x^{2}, \quad y=x+6\) (a) the \(x\) -axis (b) the line \(y=3\)
Problem 10
Find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y\). $$ \begin{array}{l} y=x^{2} \\ y=6-x \end{array} $$
Problem 11
In Exercises 11 and 12, determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=x+1, \quad g(x)=(x-1)^{2}\) (a) -2 (b) 2 (c) 10 (d) 4 (e) 8
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 11
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. $$ \frac{d y}{d x}=x(6-y) $$
Problem 11
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\). $$ y=x, \quad y=3, \quad x=0 $$
