Problem 24
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{x}, \quad y=-\frac{1}{2} x+4, \quad x=0, \quad x=8 $$
Problem 24
Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x^{2} y $$
Problem 24
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.) \(\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x\) (a) 3 (b) -2 (c) 4 (d) \(\frac{4 \pi}{3}\) (e) 1
Problem 25
Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ \({ }^{226} \mathrm{Ra} \quad 1599 \quad 10 \mathrm{~g}\)
Problem 25
Approximation In Exercises 25 and 26, approximate the arc length of the graph of the function over the interval [0,4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4 .\) Find the sum of the four lengths. (c) Use Simpson's Rule with \(n=10\) to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated are length. $$ f(x)=x^{3} $$
Problem 25
Hydraulic Press In Exercises 25 and \(26,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ \begin{array}{ll} \text { Force } & \text { Interval } \\ \hline F(x)=1000[1.8-\ln (x+1)] & 0 \leq x \leq 5 \end{array} $$
Problem 25
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=3(2-x), \quad y=0, \quad x=0 $$
Problem 26
Hydraulic Press In Exercises 25 and \(26,\) use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ F(x)=\frac{e^{x^{3}}-1}{100} \quad 0 \leq x \leq 4 $$
Problem 26
Approximation In Exercises 25 and \(26,\) approximate the arc length of the graph of the function over the interval [0,4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4\). Find the sum of the four lengths. (c) Use Simpson's Rule with \(n=10\) to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. $$ f(x)=\left(x^{2}-4\right)^{2} $$
Problem 26
Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{226} \mathrm{Ra} \quad 1599 \quad \quad 1.5 \mathrm{~g} $$
