Problem 20
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 \(x\) -axis. $$ y=\sqrt{x+2}, \quad y=x, \quad y=0 $$
Problem 20
In Exercises \(15-22,\) (a) graph the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=\ln x, \quad 1 \leq x \leq 5 $$
Problem 20
Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(x)=\sqrt[3]{x-1}, g(x)=x-1 $$
Problem 21
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=e^{-x}, \quad y=0, \quad x=0, \quad x=1 $$
Problem 21
Describe what the values of \(C\) and \(k\) represent in the exponential growth and decay model, \(y=C e^{k t}\).
Problem 21
In Exercises \(21-24,\) use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=x^{2}, \quad y=4 x-x^{2}, \text { about the line } x=4 $$
Problem 21
In Exercises \(15-22,\) (a) graph the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=2 \arctan x, \quad 0 \leq x \leq 1 $$
Problem 22
In Exercises \(15-22,\) (a) graph the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ x=\sqrt{36-y^{2}}, \quad 0 \leq y \leq 3 $$
Problem 22
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=e^{x / 2}, \quad y=0, \quad x=0, \quad x=4 $$
Problem 22
Give the differential equation that models exponential growth and decay.
