Chapter 6

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

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 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.

Find the length of the indicated curve. \(y=4 x^{3 / 2}\) between \(x=1 / 3\) and \(x=5\)

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 ; \text { about the } y \text { -axis }

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

John and Mary, weighing 180 and 110 pounds, respectively, 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?

Find the length of the indicated curve. \(y=\frac{2}{3}\left(x^{2}+1\right)^{3 / 2}\) between \(x=1\) and \(x=2\)

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 ; \text { about the } y \text { -axis }

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

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 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.)

Find the length of the indicated curve. \(y=\left(4-x^{2 / 3}\right)^{3 / 2}\) between \(x=1\) and \(x=8\)

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 PDF for a continuous random variable \(X\) is given. Use the PDF to find (a) \(P(X \geq 2),(b) E(X)\), and \((c)\) the \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} (8-x) / 32, & \text { if } 0 \leq x \leq 8 \\ 0, & \text { otherwise } \end{array}\right. $$

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

Find the length of the indicated curve. \(y=\left(x^{4}+3\right) /(6 x)\) between \(x=1\) and \(x=3\)

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 \((b) E(X)\). $$ \begin{array}{l|llll} x_{i} & 1 & 2 & 3 & 4 \\ \hline p_{i} & 0.4 & 0.2 & 0.2 & 0.2 \end{array} $$

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.

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

Find the length of the indicated curve. \(x=y^{4} / 16+1 /\left(2 y^{2}\right)\) between \(y=-3\) and \(y=-2\) Hint: Watch signs; \(\sqrt{u^{2}}=-u\) when \(u<0\).

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 \(x=5\)

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. \(y=\frac{x^{2}}{\pi}, x=4, y=0\)

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

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.

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?

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 line \(x=3\)

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. \(y=x^{3}, x=3, y=0\)

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

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?

Find the length of the indicated curve. \(y=\cosh x\), between \(x=0\) and \(x=4\)

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

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

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

Find the length of the indicated curve. \(x=e^{t} \cos t, y=e^{t} \sin t\) for \(0 \leq t \leq 1\)

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

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. \(y=e^{x}, y=\frac{e}{x}, y=0\), between \(x=0\) and \(x=3\)

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

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

Sketch the graph of the given parametric equation and find its length. \(x=t^{3} / 3, y=t^{2} / 2 ; 0 \leq t \leq 1\)

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

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

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

Sketch the graph of the given parametric equation and find its length. \(x=3 t^{2}+2, y=2 t^{3}-1 / 2 ; 1 \leq t \leq 4\)

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

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. \(y=x^{2 / 3}, y=0\), between \(x=1\) and \(x=27\)

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

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