Chapter 6

Sketch the graph of the given parametric equation and find its length. \(x=4 \sin t, y=4 \cos t-5 ; 0 \leq t \leq \pi\)

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 \(y=2\)

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. \(x=y^{2}, x=0, y=3\)

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 \(x=3\)

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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{3}{4000} x(20-x), & \text { if } 0 \leq x \leq 20 \\ 0, & \text { otherwise } \end{array}\right. $$

Find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. $$ y=\frac{1}{2}\left(x^{2}-10\right), y=0, \text { and between } x=-2 \text { and } x=2 $$

Sketch the graph of the given parametric equation and find its length. \(x=\sqrt{5} \sin 2 t-2, y=\sqrt{5} \cos 2 t-\sqrt{3} ; 0 \leq t \leq \pi / 4\)

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. \(x=\sqrt{2 y}+1, y=2, x=0, y=0 ;\) about the line \(y=3\)

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. \(x=\frac{2}{y}, y=2, y=6, x=0\)

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 \(x=3\)

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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{3}{64} x^{2}(4-x), & \text { if } 0 \leq x \leq 4 \\ 0, & \text { otherwise } \end{array}\right. $$

Find the work done in pumping all the oil (density \(\delta=50\) pounds per cubic foot) over the edge of a cylindrical tank that stands on one of its bases. Assume that the radius of the base is 4 feet, the height is 10 feet, and the tank is full of oil.

Use an \(x\) -integration to find the length of the segment of the line \(y=2 x+3\) between \(x=1\) and \(x=3 .\) Check by using the distance formula.

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. \(x=2 \sqrt{y}, y=4, x=0\)

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. \(y=(x-4)(x+2), y=0\), between \(x=0\) and \(x=3\)

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

Use a \(y\) -integration to find the length of the segment of the line \(2 y-2 x+3=0\) between \(y=1\) and \(y=3 .\) Check by using the distance formula.

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

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. \(y=x^{2}-4 x-5, y=0\), between \(x=-1\) and \(x=4\)

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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{\pi}{8} \sin (\pi x / 4), & \text { if } 0 \leq x \leq 4 \\ 0, & \text { otherwise } \end{array}\right. $$

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

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=t, y=e^{-t} ; 0 \leq t \leq 2\)

A volume \(v\) of gas is confined in a cylinder, one end of which is closed by a movable piston. If \(A\) is the area in square inches of the face of the piston and \(x\) is the distance in inches from the cylinder head to the piston, then \(v=A x .\) The pressure of the confined gas is a continuous function \(p\) of the volume, and \(p(v)=p(A x)\) will be denoted by \(f(x)\). Show that the work done by the piston in compressing the gas from a volume \(v_{1}=A x_{1}\) to a volume \(v_{2}=A x_{2}\) is $$ W=A \int_{x_{2}}^{x_{1}} f(x) d x $$

15\. Sketch the region \(R\) bounded by \(y=1 / x^{3}, x=1, x=3\), and \(y=0 .\) Set up (but do not evaluate) integrals for each of the following. (a) Area of \(R\) (b) Volume of the solid obtained when \(R\) is revolved about the \(y\) -axis (c) Volume of the solid obtained when \(R\) is revolved about \(y=-1\) (d) Volume of the solid obtained when \(R\) is revolved about \(x=4\)

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. \(x=y^{3 / 2}, y=9, x=0\)

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. \(y=\frac{1}{4}\left(x^{2}-7\right), y=0\), between \(x=0\) and \(x=2\)

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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{\pi}{8} \cos (\pi x / 8), & \text { if } 0 \leq x \leq 4 \\ 0, & \text { otherwise } \end{array}\right. $$

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=t^{2}, y=\sqrt{t} ; 1 \leq t \leq 4\)

A cylinder and piston, whose cross-sectional area is 1 square inch, contain 16 cubic inches of gas under a pressure of 40 pounds per square inch. If the pressure and the volume of the gas are related adiabatically (i.e., without loss of heat) by the law \(p v^{1.4}=c\) (a constant), how much work is done by the piston in compressing the gas to 2 cubic inches?

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. \(x=\sqrt{4-y^{2}}, x=0\)

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. \(y=x^{3}, y=0\), between \(x=-3\) 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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{4}{3} x^{-2}, & \text { if } 1 \leq x \leq 4 \\ 0, & \text { otherwise } \end{array}\right. $$

Find the volume of the solid generated by revolving the region \(R\) in the first quadrant bounded by \(y=\sqrt{-\ln x}\) and \(y=1\) about the \(x\) -axis.

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=\sin t, y=\cos 2 t ; 0 \leq t \leq \pi / 2\)

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. \(y=\sqrt[3]{x}, y=0\), between \(x=-2\) and \(x=2\)

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 \(\mathrm{CDF}\). $$ f(x)=\left\\{\begin{array}{ll} \frac{81}{40} x^{-3}, & \text { if } 1 \leq x \leq 9 \\ 0, & \text { otherwise } \end{array}\right. $$

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=t \ln t, y=t-1 ; 1 \leq t \leq 3\)

One cubic foot of gas under a pressure of 80 pounds per square inch expands adiabatically to 4 cubic feet according to the law \(p v^{1.4}=c\). Find the work done by the gas.

Find the volume of the solid generated by revolving about the \(x\) -axis the region bounded by the line \(y=6 x\) and the parabola \(y=6 x^{2}\).

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. \(y=\sqrt{x}-10, y=0\), between \(x=0\) and \(x=9\)

A round hole of radius \(a\) is drilled through the center of a solid sphere of radius \(b\) (assume that \(b>a\) ). Find the volume of the solid that remains.

Sketch the graph of the four-cusped hypocycloid \(x=a \sin ^{3} t, y=a \cos ^{3} t, 0 \leq t \leq 2 \pi\), and find its length. Hint: By symmetry, you can quadruple the length of the first quadrant portion.

A cable weighing 2 pounds per foot is used to haul a 200 pound load to the top of a shaft that is 500 feet deep. How much work is done?

Find the volume of the solid generated by revolving about the \(x\) -axis the region bounded by the line \(x-2 y=0\) and the parabola \(y^{2}=4 x\)

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. \(y=(x-3)(x-1), y=x\)

A continuous random variable \(X\) is said to have a uniform distribution on the interval \([a, b]\) if the PDF has the form $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{b-a}, & \text { if } a \leq x \leq b \\ 0, & \text { otherwise } \end{array}\right. $$ (a) Find the probability that the value of \(X\) is closer to \(a\) than it is to \(b\). (b) Find the expected value of \(X\). (c) Find the CDF of \(X\).

Set up the integral (using shells) for the volume of the torus obtained by revolving the region inside the circle \(x^{2}+y^{2}=a^{2}\) about the line \(x=b\), where \(b>a\). Then evaluate this integral. Hint: As you simplify, it may help to think of part of this integral as an area.

A 10 -pound monkey hangs at the end of a 20 -foot chain that weighs \(\frac{1}{2}\) pound per foot. How much work does it do in climbing the chain to the top? Assume that the end of the chain is attached to the monkey.

Find the volume of the solid generated by revolving about the \(x\) -axis the region in the first quadrant bounded by the circle \(x^{2}+y^{2}=r^{2}\), the \(x\) -axis, and the line \(x=r-h, 0

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. \(y=\sqrt{x}, y=x-4, x=0\)