Chapter 27

Find the area bounded by the curves \(y=e^{x}, y=1-x\), and \(x=1\).

A cylinder 80 centimeters tall with a 10 -centimeter radius is filled with a compressible substance. The density of this substance is given by \(\rho(h)\) grams per cubic centimeter, where \(h\) is the height (in centimeters) from the bottom of the cylinder. Write an expression for the total mass of the substance in the cylinder.

Find the area bounded by \(y=2-x^{2}\) and \(y=x\).

A city is in the shape of a rectangle 4 miles wide by 6 miles long. A river runs through the middle of the city, parallel to the 6 mile-long sides. People prefer to live nearer the water, so the density of people is given by \(\rho(x)=10,000-800 x\) people per square mile, where \(x\) is the distance from the river. (You may ignore the width of the river in this problem.) (a) Show in a sketch how you will need to slice up the region. (b) What is the area of the \(i\) th slice? (c) What is the approximate population in the \(i\) th slice? (d) Write a Riemann sum to estimate the total population of the city. (e) Calculate the exact population by taking the limit of the Riemann sum and evaluating the resulting definite integral.

Find the area bounded above by \(y=-x+6\), below by \(y=x^{2}+1\), and on the left by the \(y\) -axis.

Traditionally, when a college football team seems certain to receive a bid to play in the post-season Orange Bowl, fans begin to throw oranges onto the field. Suppose that at one point during a game, the number of oranges per square yard between the goal line and the 30 -yard line is given by \(\rho(x)=\frac{30-x}{3}\) oranges per square yard, where \(x\) is the number of yards from the goal line. If the field is 160 feet ( \(160 / 3\) yards) wide, how many oranges lie between the goal line and the 30 -yard line?

Find the area bounded below by the \(x\) -axis, on the left by the \(y\) -axis, and above by \(y=x^{2}+1\) and \(y=-x+6\).

(a) A farmer has planted corn on a rectangular plot of land 800 meters by 1000 meters. A straight stream runs alongside one of the long borders of the plot, and the farmer's irrigation system is such that his yield decreases with the distance from the stream. Suppose his yield is given by \(f(x)=50-0.3 \sqrt{x}\) ears of corn per square meter, where \(x\) is the distance from the stream in meters. What is the farmer's yield from the plot? (b) A second farmer plants his corn in a circular plot with radius 80 meters and he has a centralized irrigation system located in the middle of his field. His yield drops with the distance from the center of the field. Suppose his yield is also given by \(f(x)=50-0.3 \sqrt{x}\) ears of corn per square meter, this time \(x\) being the distance from the center of the field. What is the farmer's yield from this plot?

Find the area in the first quadrant bounded by \(y=\arcsin x, y=\pi / 2\), and \(x=0 .\) (Hint: To get an exact answer it will be simplest to integrate with respect to \(y .\).)

Consider a box of cereal with raisins. The box is 5 centimeters deep, 25 centimeters tall, and 16 centimeters wide. The raisins tend to fall toward the bottom; assume their density is given by \(\rho(h)=\frac{4}{h+10}\) raisins per cubic centimeter, where \(h\) is the height above the bottom of the box. How many raisins are in the box?

(a) Let \(A\) be the area between the cosine and the sine curves between \(x=-\pi / 4\) and \(x=\pi / 4\) i. Sketch the graphs of \(\sin x\) and \(\cos x\), shading the region \(A\) described above. ii. Write a definite integral giving the area \(A\). (b) Let \(B\) be the area between the cosine and the sine curves between \(x=0\) and \(x=2 \pi\) i. Sketch the graphs of \(\sin x\) and \(\cos x\), shading the region \(B\) described above. ii. Write a sum of the definite integrals giving the area \(B\). Notice that over a certain interval the sine curve lies above the cosine curve, while on other intervals it lies below it. Therefore, it is necessary to split up the interval and write a sum of integrals.

The density of dart holes on an old dartboard is given by \(\rho(r)=\frac{1010}{\pi\left(r^{2}+1\right)^{2}}\) holes per square inch, where \(r\) is the distance, in inches, from the center of the board. If the board is a circle with diameter 20 inches, find the total number of holes in the board.

A beam of light is shining onto a screen creating a disk of radius 50 centimeters. The intensity of light is brightest at the center and diminishes away from the center. If the intensity of light at a distance \(r\) from the center of the beam is given by \(f(r)=\frac{150}{20+r^{2}}\) watt/square \(\mathrm{cm}\), find the total wattage of the beam's image on the screen.

A coastal town is in the shape of a 7 -mile by 2 -mile rectangle, with one of the 7 -mile sides along the coast. In this town people want to live near the beach and the population density at a distance \(x\) from the coast is given by \(\delta(x)=4000-2000 x\) people per square mile. (a) Write a general Riemann sum that approximates the total population of the town. (b) Use your answer to part (a) to write a definite integral that represents the total population of the town and evaluate the integral.

Find the area between the curve \(y=\ln x\) and the \(x\) -axis for \(1 \leq x \leq 10\). Get an exact answer. (Hint: Slice the area perpendicular to the \(y\) -axis so that the height of each slice is \(\Delta y\). Use this to arrive at an integral that you can evaluate exactly.)

Find (exactly) the area bounded by \(x=1 / e, y=\ln x\), and \(y=1\).

(a) Suppose that the density of organisms in a certain petridish varies with the distance from the center of the dish. The density at a distance \(x\) centimeters from the center is given by \(f(x)\) organisms per square centimeter. The petri dish is 18 centimeters in diameter. i. Write an integral that gives the number of organisms in the dish. ii. Find the number of organisms in the dish if \(f(x)=100 e^{-x^{2}}\) organisms per square centimeter. (b) Suppose that the density of organisms in a certain petri dish varies with the distance from a strip of nutrients running along the diameter of the dish. The density at a distance \(x\) centimeters from the line of nutrients is given by \(f(x)\) organisms per square centimeter. The petri dish is 18 centimeters in diameter. i. How will you slice up the petri dish? ii. Approximate the number of organisms in the \(i\) th slice. iii. Write a Riemann sum approximating the total number of organisms in the petri dish. iv. Write an integral that gives the number of organisms in the dish.

Find the area bounded below by the \(x\) -axis, and laterally by \(y=\ln x\), and the line segment joining \((e, 1)\) to \((2 e, 0)\).

Suppose that the density of a planet of mass in a gaseous planet is given by the function \(\rho(r)=\frac{40000}{1+.0001 r^{3}}\) kilograms per cubic kilometer, where \(r\) is the number of kilometers from the center of the planet. Find the total mass of the planet if it has a radius of 8000 kilometers.

Evaluate \(\int_{0}^{1} \arctan x d x\) by interpreting it as an area and slicing horizontally.

A chocolate truffle is a wonderfully decadent chocolate concoction. Truffles tend to be spherical or hemispherical. (a) Consider a truffle made by dipping a round hazelnut into various chocolates, building up a delicious spherical delicacy. The number of calories per cubic millimeter varies with \(x\), where \(x\) is the distance from the center of the hazelnut. If \(\rho(x)\) gives the calories \(/ \mathrm{mm}^{3}\) at a distance \(x\) millimeters from the center, write an integral that gives the number of calories in a truffle of radius \(R\). (b) Another truffle is made in a hemispherical mold with radius \(R\). Layers of different types of chocolate are poured into the mold, one at a time, and allowed to set. The number of calories per cubic millimeter varies with \(x\), where \(x\) is the depth from the top of the mold. The calorie density is given by \(\delta(x)\) calories \(/ \mathrm{mm}^{3}\). Write an integral that gives the number of calories in this hemispherical truffle.

Evaluate \(\int_{0}^{0.5} \arcsin x d x\)

Liquid is being stored in a large spherical tank of radius 2 meters. The tank is completely full and has been left standing for a long time. A mineral suspended in the liquid is setting. Its density at a depth of \(h\) meters from the top is given by \(5 h\) milligrams per cubic meter. Determine the number of milligrams of the mineral contained in the tank.

The region \(A\) in the first quadrant is bounded by \(y=2 x, y=-3 x+10\), and \(y=-\frac{1}{9}\left(x^{2}-6 x\right) .\) It has corners at \((0,0),(2,4)\), and \((3,1) .\) Express the area of \(A\) is the sum or difference of definite integrals. You need not evaluate.

Let \(W(t)\) be the amount of water in a pool at time \(t, t\) measured in hours and \(W\) measured in gallons. \(t=0\) corresponds to noon. Water is flowing in and out of the pool at a rate given by \(\frac{d W}{d t}=30 \cos \left(\frac{\pi}{2} t\right)\). During what time interval between noon and \(5: 00\) P.M. \((0 \leq t \leq 5)\) is water flowing out of the pool at a rate of 15 gallons an hour or more? How much water actually has left the pool in this time interval?

In the town of Lybonrehc there has been a nuclear reactor meltdown, which released radioactive iodine \(131 .\) Fortunately, the reactor has a containment building, which kept the iodine from being released into the air. The containment building is hemispherical with a radius of 100 feet. The density of iodine in the building was \(6 \times 10^{-5}(200-h)\) \(\mathrm{g} /\) cubic feet, where \(h\) is the height from the floor (in feet). (It ranges from \(12 \times 10^{-3}\) g/cubic feet at the floor to \(6 \times 10^{-3} \mathrm{~g} /\) cubic feet near the top. \()\) (a) Derive an integral that gives the amount of iodine in the building. You must explain your reasoning fully and clearly. (b) Calculate the amount of iodine in the building.

A spherical star has a radius of 90,000 kilometers. The density of matter in the star is given by \(\rho(r)=\frac{\mathrm{K}}{(r+1)^{3 / 2}}\) kilograms per cubic kilometer, where \(r\) is the distance (in kilometers) from the star's center and \(\mathrm{K}\) is a positive constant. Write out (but do not evaluate) an expression for the total mass of the star. Your answer should contain the constant \(\mathrm{K}\).

A substance has been put in a centrifuge. We now have a cylindrical sample (radius 3 centimeters, height 4 centimeters) in which density varies with \(x\), the distance (in centimeters) from the central axis. If the density is given by \(\rho(x) \mathrm{mg} / \mathrm{cm}^{3}\), write an integral that gives the total mass of the substance.

A circus tent has cylindrical symmetry about its center pole. The height a distance of \(x\) feet from the center pole is given by \(h(x)=\frac{8}{1+\frac{x^{2}}{16}}\) feet. What is the volume enclosed by the tent of radius \(4 ?\)

(a) What is the present value of a single payment of \(\$ 2000\) three years in the future? Assume \(5 \%\) interest compounded continuously. (b) What is the present value of a continuous stream of income at the rate of \(\$ 100,000\) per year over the next 20 years? Assume \(5 \%\) interest compounded continuously. By "a continuous stream of income" we mean that we are modeling the situation by assuming that money is being generated continuously at a rate of \(\$ 100,000\) per year. Begin by partitioning the time interval \([0,20]\) into \(n\) equal pieces. Figure out the amount of money generated in the \(i\) th interval and pull it back to the present. Summing these pull-backs should approximate the present value of the entire income stream.