Chapter 6

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=\frac{1}{x}, x=1, x=4, y=0 ;\) about the \(y\)-axis

y=4 x^{3 / 2} \text { between } x=1 / 3 \text { and } x=5

Particles of mass \(m_{1}=5, m_{2}=7\), and \(m_{3}=9\) are located at \(x_{1}=2, x_{2}=-2\), and \(x_{3}=1\) along a line. Where is the center of mass?

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). \begin{array}{l|llll} x_{i} & 0 & 1 & 2 & 3 \\ \hline p_{i} & 0.80 & 0.10 & 0.05 & 0.05 \end{array}

A force of 6 pounds is required to keep a spring stretched \(\frac{1}{2}\) foot beyond its normal length. Find the value of the spring constant and the work done in stretching the spring \(\frac{1}{2}\) foot beyond its natural length.

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=x^{2}, x=1, y=0 ;\) about the \(y\)-axis

John and Mary, weighing 180 and 110 pounds, respectiveIy, sit at opposite ends of a 12 -foot teeter board with the fulcrum in the middle. Where should their 80 -pound son Tom sit in order for the board to balance?

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ \begin{array}{l|lllll} x_{i} & 0 & 1 & 2 & 3 & 4 \\ \hline p_{i} & 0.70 & 0.15 & 0.05 & 0.05 & 0.05 \end{array} $$

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=\sqrt{x}, x=3, y=0\); about the \(y\)-axis

A straight wire 7 units long has density \(\delta(x)=\sqrt{x}\) at a point \(x\) units from one end. Find the distance from this end to the center of mass.

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ \begin{array}{l|lllll} x_{i} & -2 & -1 & 0 & 1 & 2 \\ \hline p_{i} & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \end{array} $$

A force of \(0.6\) newton is required to keep a spring with a natural length of \(0.08\) meter compressed to a length of \(0.07\) meter. Find the work done in compressing the spring from its natural length to a length of \(0.06\) meter. (Hooke's Law applies to compressing as well as stretching.)

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=9-x^{2}(x \geq 0), x=0, y=0 ;\) about the \(y\)-axis

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ \begin{array}{l|lllll} x_{i} & -2 & -1 & 0 & 1 & 2 \\ \hline p_{i} & 0.1 & 0.2 & 0.4 & 0.2 & 0.1 \end{array} $$

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=\sqrt{x}, x=5, y=0 ;\) about the line \(x=5\)

The masses and coordinates of a system of particles in the coordinate plane are given by the following: \(2,(1,1) ; 3,(7,1)\); \(4,(-2,-5) ; 6,(-1,0) ; 2,(4,6)\). Find the moments of this system with respect to the coordinate axes, and find the coordinates of the center of mass.

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ \begin{array}{l|llll} x_{i} & 1 & 2 & 3 & 4 \\ \hline p_{i} & 0.4 & 0.2 & 0.2 & 0.2 \end{array} $$

In Problems 5-10, sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\)-axis. $$ y=\frac{x^{2}}{\pi}, x=4, y=0 $$

For any spring obeying Hooke's Law, show that the work done in stretching a spring a distance \(d\) is given by \(W=\frac{1}{2} k d^{2}\).

30 x y^{3}-y^{8}=15 \text { between } y=1 \text { and } y=3

The masses and coordinates of a system of particles are given by the following: \(5,(-3,2) ; 6,(-2,-2) ; 2,(3,5) ; 7,(4,3)\); \(1,(7,-1)\). Find the moments of this system with respect to the coordinate axes, and find the coordinates of the center of mass.

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ \begin{array}{l|lll} x_{i} & -0.1 & 100 & 1000 \\ \hline p_{i} & 0.980 & 0.018 & 0.002 \end{array} $$

In Problems 5-10, sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\)-axis. $$ y=x^{3}, x=3, y=0 $$

For a certain type of nonlinear spring, the force required to keep the spring stretched a distance \(s\) is given by the formula \(F=k s^{4 / 3}\). If the force required to keep it stretched 8 inches is 2 pounds, how much work is done in stretching this spring 27 inches?

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=\frac{1}{4} x^{3}+1, y=1-x, x=1 ;\) about the \(y\)-axis

y=\cosh x, \text { between } x=0 \text { and } x=4

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ p_{i}=(5-i) / 10, x_{i}=i, i=1,2,3,4 $$

In Problems 5-10, sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\)-axis. $$ y=\frac{1}{x}, x=2, x=4, y=0 $$

A spring is such that the force required to keep it stretched \(s\) feet is given by \(F=9 s\) pounds. How much work is done in stretching it 2 feet?

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(y=x^{2}, y=3 x\); about the \(y\)-axis

Find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. \(y=2-x, y=0, x=0\)

A discrete probability distribution for a random variable \(X\) is given. Use the given distribution to find (a) \(P(X \geq 2)\) and (b) \(E(X)\). $$ p_{i}=(2-i)^{2} / 10, x_{i}=i, i=0,1,2,3,4 $$

In Problems 5-10, sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\)-axis. $$ y=e^{x}, y=\frac{e}{x}, y=0, \text { between } x=0 \text { and } x=3 $$

Two similar springs \(S_{1}\) and \(S_{2}\), each 3 feet long, are such that the force required to keep either of them stretched a distance of \(s\) feet is \(F=6 s\) pounds. One end of one spring is fastened to an end of the other, and the combination is stretched between the walls of a room 10 feet wide (Figure 17). What work is done in moving the midpoint, \(P, 1\) foot to the right?

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=y^{2}, y=1, x=0 ;\) about the \(x\)-axis

Find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. \(y=2-x^{2}, y=0\)

A PDF for a continuous random variable \(X\) is given. Use the \(P D F\) to find (a) \(P(X \geq 2)\), (b) \(E(X)\), and (c) the CDF: $$ f(x)= \begin{cases}\frac{1}{20}, & \text { if } 0 \leq x \leq 20 \\ 0, & \text { otherwise }\end{cases} $$

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=\sqrt{y}+1, y=4, x=0, y=0 ;\) about the \(x\)-axis

x=3 t^{2}+2, y=2 t^{3}-1 / 2 ; 1 \leq t \leq 4

Find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. \(y=\frac{1}{3} x^{2}, y=0, x=4\)

A PDF for a continuous random variable \(X\) is given. Use the \(P D F\) to find (a) \(P(X \geq 2)\), (b) \(E(X)\), and (c) the CDF: $$ f(x)= \begin{cases}\frac{1}{40}, & \text { if }-20 \leq x \leq 20 \\ 0, & \text { otherwise }\end{cases} $$

In Problems 5-10, sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\)-axis. $$ y=x^{2 / 3}, y=0, \text { between } x=1 \text { and } x=27 $$

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=y^{2}, y=2, x=0 ;\) about the line \(y=2\)

In Problems 11-30, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer: \(y=3-\frac{1}{3} x^{2}, y=0\), between \(x=0\) and \(x=3\)

x=4 \sin t, y=4 \cos t-5 ; 0 \leq t \leq \pi

Find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. \(y=x^{3}, y=0, x=1\)

A PDF for a continuous random variable \(X\) is given. Use the \(P D F\) to find (a) \(P(X \geq 2)\), (b) \(E(X)\), and (c) the CDF: $$ f(x)= \begin{cases}\frac{3}{256} x(8-x), & \text { if } 0 \leq x \leq 8 \\ 0, & \text { otherwise }\end{cases} $$

In Problems 11-16, sketch the region \(R\) bounded by the graphs of the given equations and show a typical horizontal slice. Find the volume of the solid generated by revolving \(R\) about the \(y\)-axis. $$ x=y^{2}, x=0, y=3 $$

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=\sqrt{2 y}+1, y=2, x=0, y=0 ;\) about the line \(y=3\)

In Problems 11-30, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer: \(y=5 x-x^{2}, y=0\), between \(x=1\) and \(x=3\)