Chapter 6

A space capsule weighing 5000 pounds is propelled to an altitude of 200 miles above the surface of the earth. How much work is done against the force of gravity? Assume that the earth is a sphere of radius 4000 miles and that the force of gravity is \(f(x)=-k / x^{2}\), where \(x\) is the distance from the center of the earth to the capsule (the inverse-square law). Thus, the lifting force required is \(k / x^{2}\), and this equals 5000 when \(x=4000\).

The region bounded by \(y=2+\sin x, y=0, x=0\), and \(x=2 \pi\) is revolved about the \(y\)-axis. Find the volume that results. Hint: \(\int x \sin x d x=\sin x-x \cos x+C\).

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: $$ y=x^{2}-9, y=(2 x-1)(x+3) $$

Without doing any integration, find the median of the random variable that has \(\mathrm{PDF} \quad f(x)=\frac{15}{512} x^{2}(4-x)^{2}\), \(0 \leq x \leq 4\).

Find the volume of the solid generated by revolving about the line \(y=2\) the region in the first quadrant bounded by the parabolas \(3 x^{2}-16 y+48=0\) and \(x^{2}-16 y+80=0\) and the \(y\)-axis.

According to Coulomb's Law, two like electrical charges repel each other with a force that is inversely proportional to the square of the distance between them. If the force of repulsion is 10 dynes ( 1 dyne \(=10^{-5}\) newton ) when they are 2 centimeters apart, find the work done in bringing the charges from 5 centimeters apart to 1 centimeter apart.

Let \(R\) be the region bounded by \(y=x^{2}\) and \(y=x\). Find the volume of the solid that results when \(R\) is revolved about: (a) the \(x\)-axis; (b) the \(y\)-axis; (c) the line \(y=x\).

23\. Find the length of each curve. (a) \(y=\int_{1}^{x} \sqrt{u^{3}-1} d u, 1 \leq x \leq 2\) (b) \(x=t-\sin t, y=1-\cos t, 0 \leq t \leq 4 \pi\)

Find the value of \(k\) that makes \(f(x)=k x(5-x)\), \(0 \leq x \leq 5\), a valid PDF. Hint: The PDF must integrate to 1 .

The base of a solid is the region inside the circle \(x^{2}+y^{2}=4\). Find the volume of the solid if every cross section by a plane perpendicular to the \(x\)-axis is a square. Hint: See Examples 5 and 6 .

A bucket weighing 100 pounds is filled with sand weighing 500 pounds. A crane lifts the bucket from the ground to a point 80 feet in the air at a rate of 2 feet per second, but sand simultaneously leaks out through a hole at 3 pounds per second. Neglecting friction and the weight of the cable, determine how much work is done. Hint: Begin by estimating \(\Delta W\), the work required to lift the bucket from \(y\) to \(y+\Delta y\).

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: $$ x=(3-y)(y+1), x=0 $$

Find the length of each curve. (a) \(y=\int_{\pi / 6}^{x} \sqrt{64 \sin ^{2} u \cos ^{4} u-1} d u, \frac{\pi}{6} \leq x \leq \frac{\pi}{3}\) (b) \(x=a \cos t+a t \sin t, y=a \sin t-a t \cos t,-1 \leq t \leq 1\)

Find the value of \(k\) that makes \(f(x)=k x^{2}(5-x)^{2}\), \(0 \leq x \leq 5\), a valid PDF.

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: $$ x=-6 y^{2}+4 y, x+3 y-2=0 $$

The time in minutes that it takes a worker to complete a task is a random variable with PDF \(f(x)=k(2-|x-2|)\), \(0 \leq x \leq 4\). (a) Find the value of \(k\) that makes this a valid PDF. (b) What is the probability that it takes more than 3 minutes to complete the task? (c) Find the expected value of the time to complete the task. (d) Find the CDF \(F(x)\). (e) Let \(Y\) denote the time in seconds required to complete the task. What is the CDF of \(Y\) ? Hint: \(P(Y \leq y)=\) \(P(60 X \leq y)\)

The base of a solid is bounded by one arch of \(y=\sqrt{\cos x},-\pi / 2 \leq x \leq \pi / 2\), and the \(x\)-axis. Each cross section perpendicular to the \(x\)-axis is a square sitting on this base. Find the volume of the 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: $$ x=y^{2}-2 y, x-y-4=0 $$

y=\sqrt{25-x^{2}},-2 \leq x \leq 3

Use Pappus's Theorem to find the volume of the torus obtained when the region inside the circle \(x^{2}+y^{2}=a^{2}\) is revolved about the line \(x=2 a\).

The daily summer air quality index \((\mathrm{AQI})\) in \(\mathrm{St}\). Louis is a random variable whose \(\mathrm{PDF}\) is \(f(x)=k x^{2}(180-x)\), \(0 \leq x \leq 180 .\) (a) Find the value of \(k\) that makes this a valid PDF. (b) A day is an "orange alert" day if the AQI is between 100 and 150 . What is the probability that a summer day is an orange alert day? (c) Find the expected value of the summer AQI.

The base of a solid is the region bounded by \(y=1-x^{2}\) and \(y=1-x^{4}\). Cross sections of the solid that are perpendicular to the \(x\)-axis are squares. Find the volume of the 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: $$ 4 y^{2}-2 x=0,4 y^{2}+4 x-12=0 $$

Use Pappus's Theorem together with the known volume of a sphere to find the centroid of a semicircular region of radius \(a\).

Holes drilled by a machine have a diameter, measured in millimeters, that is a random variable with PDF \(f(x)=\) \(k x^{6}(0.6-x)^{8}, 0 \leq x \leq 0.6\) (a) Find the value of \(k\) that makes this a valid PDF. (b) Specifications require that the hole's diameter be between \(0.35\) and \(0.45 \mathrm{~mm}\). Those units not meeting this requirement are scrapped. What is the probability that a unit is scrapped? (c) Find the expected value of the hole's diameter. (d) Find the CDF \(F(x)\). (e) Let \(Y\) denote the hole's diameter in inches. (1 inch = \(25.4 \mathrm{~mm}\).) What is the CDF of \(Y\) ?

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: $$ x=4 y^{4}, x=8-4 y^{4} $$

\text { 28. } y=\left(x^{6}+2\right) /\left(8 x^{2}\right), 1 \leq x \leq 3

A company monitors the total impurities in incoming batches of chemicals. The PDF for total impurity \(X\) in a batch, measured in parts per million (PPM), has PDF \(f(x)=\) \(k x^{2}(200-x)^{8}, 0 \leq x \leq 200 .\) (a) Find the value of \(k\) that makes this a valid PDF. (b) The company does not accept batches whose total impurity is 100 or above. What is the probability that a batch is not accepted? (c) Find the expected value of the total impurity in PPM. (d) Find the CDF \(F(x)\). (e) Let \(Y\) denote the total impurity in percent, rather than in PPM. What is the CDF of \(Y\) ?

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: \(y=e^{2 x}, y=0\), between \(x=0\) and \(x=\ln 2\)

Suppose that \(X\) is a random variable that has a uniform distribution on the interval \([0,1]\). (See Problem 20.) The point \((1, X)\) is plotted in the plane. Let \(Y\) be the distance from \((1, X)\) to the origin. Find the CDF and the PDF of the random variable \(Y\).

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

x=1-t^{2}, y=2 t, 0 \leq t \leq 1

Suppose that \(X\) is a continuous random variable. Explain why \(P(X=x)=0\). Which of the following probabilities are the same? Explain. $$ \begin{aligned} &P(a

Sketch the region \(R\) bounded by \(y=x+6, y=x^{3}\), and \(2 y+x=0\). Then find its area. Hint: Divide \(R\) into two pieces.

y=\sqrt{r^{2}-x^{2}},-r \leq x \leq r

Find the centroid of the region bounded by \(y=e^{-x}, x=0, x=2\), and \(y=0\).

Show that if \(A^{c}\) is the complement of \(A\), that is, the set of all outcomes in the sample space \(S\) that are not in \(A\), then \(P\left(A^{c}\right)=1-P(A)\).

The base of a solid is the region \(R\) bounded by \(y=\sqrt{x}\) and \(y=x^{2}\). Each cross section perpendicular to the \(x\)-axis is a semicircle with diameter extending across \(R\). Find the volume of the solid.

Find the total force exerted by the water on all sides of a cube of edge length 2 feet if its top is horizontal and 100 feet below the surface of a lake.

An object moves along a line so that its velocity at time \(t\) is \(v(t)=3 t^{2}-24 t+36\) feet per second. Find the displacement and total distance traveled by the object for \(-1 \leq t \leq 9\).

Find the volume of the solid generated by revolving the region bounded by \(y=e^{x}, y=0, x=0\), and \(x=\ln 3\) about the \(x\)-axis.

Find the volume of the solid generated when the region in the first quadrant bounded above by \(y=2\) and on the right by \(y=-\ln x\) is revolved about the \(y\)-axis.

Find the total force exerted by the fluid against the lateral surface of a right circular cylinder of height 6 feet, which stands on its circular base of radius 5 feet, if it is filled with oil \((\delta=50\) pounds per cubic foot \()\).

Starting at \(s=0\) when \(t=0\), an object moves along a line so that its velocity at time \(t\) is \(v(t)=2 t-4\) centimeters per second. How long will it take to get to \(s=12\) ? To travel a total distance of 12 centimeters?

. If the surface of a cone of slant height \(\ell\) and base radius \(r\) is cut along a lateral edge and laid flat, it becomes the sector of a circle of radius \(\ell\) and central angle \(\theta\) (see Figure 19 ). (a) Show that \(\theta=2 \pi r / \ell\) radians. (b) Use the formula \(\frac{1}{2} \ell^{2} \theta\) for the area of a sector of radius \(\ell\) and central angle \(\theta\) to show that the lateral surface area of a cone is \(\pi r \ell\). (c) Use the result of part (b) to obtain the formula \(A=2 \pi\left[\left(r_{1}+r_{2}\right) / 2\right] \ell\) for the lateral area of a frustum of a cone with base radii \(r_{1}\) and \(r_{2}\) and slant height \(\ell\).

Let \(0 \leq f(x) \leq g(x)\) for all \(x\) in \([0,1]\), and let \(R\) and \(S\) be the regions under the graphs of \(f\) and \(g\), respectively. Prove or disprove that \(\bar{y}_{R} \leq \bar{y}_{S}\).

Suppose a random variable \(Y\) has CDF $$ F(y)= \begin{cases}0, & \text { if } y<0 \\ 2 y /(y+1), & \text { if } 0 \leq y \leq 1 \\ 1, & \text { if } y>1\end{cases} $$ Find each of the following: (a) \(P(Y<2)\) (b) \(P(0.5

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the curve \(y^{2}=x^{3}\), the line \(x=4\), and the \(x\)-axis: (a) about the line \(x=4\); (b) about the line \(y=8\).

Consider the curve \(y=1 / x^{2}\) for \(1 \leq x \leq 6\). (a) Calculate the area under this curve. (b) Determine \(c\) so that the line \(x=c\) bisects the area of part (a). (c) Determine \(d\) so that the line \(y=d\) bisects the area of part (a).

. Show that the area of the part of the surface of a sphere of radius \(a\) between two parallel planes \(h\) units apart \((h<2 a)\) is \(2 \pi a h\). Thus, show that if a right circular cylinder is circumscribed about a sphere then two planes parallel to the base of the cylinder bound regions of the same area on the sphere and the cylinder.