Chapter 6

x=\sqrt{5} \sin 2 t-2, y=\sqrt{5} \cos 2 t-\sqrt{3} ; 0 \leq t \leq \pi / 4

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

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: $$ f(x)= \begin{cases}\frac{3}{4000} x(20-x), & \text { if } 0 \leq x \leq 20 \\\ 0, & \text { otherwise }\end{cases} $$

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

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

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.

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

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: $$ f(x)= \begin{cases}\frac{3}{64} x^{2}(4-x), & \text { if } 0 \leq x \leq 4 \\\ 0, & \text { otherwise }\end{cases} $$

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

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.

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

. 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.

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

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: $$ f(x)= \begin{cases}(8-x) / 32, & \text { if } 0 \leq x \leq 8 \\ 0, & \text { otherwise }\end{cases} $$

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. $$ x=y^{2 / 3}, y=27, x=0 $$

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 \(x=4\)

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

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

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: $$ f(x)= \begin{cases}\frac{\pi}{8} \sin (\pi x / 4), & \text { if } 0 \leq x \leq 4 \\ 0, & \text { otherwise }\end{cases} $$

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. $$ x=y^{3 / 2}, y=9, x=0 $$

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 $$ W=A \int_{x_{2}}^{x_{1}} f(x) d x $$ Hint: The total force on the face of the piston is \(p(v) \cdot A=\) \(p(A x) \cdot A=A \cdot f(x) .\)

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: \(y=x^{3}, y=0\), between \(x=-3\) and \(x=3\)

x=t^{2}, y=\sqrt{t} ; 1 \leq t \leq 4

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

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: $$ f(x)= \begin{cases}\frac{\pi}{8} \cos (\pi x / 8), & \text { if } 0 \leq x \leq 4 \\ 0, & \text { otherwise }\end{cases} $$

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

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?

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

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: $$ f(x)= \begin{cases}\frac{4}{3} x^{-2}, & \text { if } 1 \leq x \leq 4 \\ 0, & \text { otherwise }\end{cases} $$

Find the volume of the solid generated by revolving about the \(x\)-axis the region bounded by the upper half of the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ and the \(x\)-axis, and thus find the volume of a prolate spheroid. Here \(a\) and \(b\) are positive constants, with \(a>b\).

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

x=t \ln t, y=t-1 ; 1 \leq t \leq 3

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: $$ f(x)= \begin{cases}\frac{81}{40} x^{-3}, & \text { if } 1 \leq x \leq 9 \\ 0, & \text { otherwise }\end{cases} $$

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 \(y=6 x^{2}\).

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.

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.

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

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.

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 \(y^{2}=4 x\).

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?

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.

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: $$ y=\sqrt{x}, y=x-4, x=0 $$

. A point \(P\) on the rim of a wheel of radius \(a\) is initially at the origin. As the wheel rolls to the right along the \(x\)-axis, \(P\) traces out a curve called a cycloid (see Figure 18). Derive parametric equations for the cycloid as follows. The parameter is \(\theta\). (a) Show that \(\overline{O T}=a \theta\). (b) Convince yourself that \(\overline{P Q}=a \sin \theta, \overline{Q C}=a \cos \theta\), \(0 \leq \theta \leq \pi / 2\).

A continuous random variable \(X\) is said to have a uniform distribution on the interval \([a, b]\) if the PDF has the form $$ f(x)= \begin{cases}\frac{1}{b-a}, & \text { if } a \leq x \leq b \\ 0, & \text { otherwise }\end{cases} $$ (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 \(X\).

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

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.

The region in the first quadrant bounded by \(x=0, y=\sin \left(x^{2}\right)\), and \(y=\cos \left(x^{2}\right)\) is revolved about the \(y\)-axis. Find the volume of the resulting solid.

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: $$ y=x^{2}-2 x, y=-x^{2} $$

The median of a continuous random variable \(X\) is a value \(x_{0}\) such that \(P\left(X \leq x_{0}\right)=0.5\). Find the median of a uniform random variable on the interval \([a, b]\).

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