Exam 8: Using the Definite Integral

Alice starts at the origin and walks along the graph of $y=\frac{x^{2}}{2}$ at a velocity of 6 units/second.Which of the following integrals show how far Alice has traveled when she reaches the point where x = a?

If a cylinder has radius 6 and height h (in centimeters), which of the following definite integrals represents the volume using slices?

In March 1995 the space shuttle carried an experiment designed by a Harvard student who studies the growth of crystals.Suppose the probability density function of the length, x cm, of a crystal grown in space is modeled by $p(x)=x e^{-x}$ for $x \geq 0$ .The cumulative distribution function giving the probability that a crystal has length $\le$ t cm is represented by P(t).Which of the quantities (i)- (x)below represents precisely the probability that a crystal has length less than 3.01 cm?

A flag is hanging over the side of a building.Assume that the flag is a right triangle that has a width (one leg) of 25 feet at the top of the building, a length (other leg)of 45 feet hanging down the side of the building, and a weight of 89 pounds.Find the amount of work (in ft-lbs)needed to lift this flag to the roof.

A thin strip of nutrients 20 cm long is placed in a circular petri dish of radius 10 cm, as shown.The population density of bacteria in the disk after 3 hours is given by $\frac{75}{D+4}$ bacteria/cm² where D is the distance (in cm)to the nutrient strip.Which of the following integrals gives the number of bacteria in the petri dish 3 hours after the nutrient strip has been introduced?

A lightbulb company is interested in the lifespan of their lightbulbs.They have 20,000 lightbulbs burning and have collected the following information. After 2 months, 98% of the bulbs were still working. After 8 months, 80% of the bulbs were still working. We summarize all the data collected below: (Read carefully: the data was not collected at regular intervals.) Approximate the average lifespan of a lightbulb in months.Round to 2 decimal places.

The density of a compressible liquid is 30(5 - h)kg/m³ at a height of h meters above the bottom.The liquid is put in the container as shown below (resting on the triangular side).The cross sections of the container are isosceles triangles, it has straight sides, and it looks like a triangular prism.How many kg will it hold? Round to the nearest whole number.

A study of the costs to produce airplanes in World War II led to the theory of "learning curves," the idea of which is that the marginal cost per plane decreases over the duration of a production run.In other words, with experience, staff on an assembly line can produce planes with greater efficiency.The 90% learning curve describes a typical situation where the marginal cost, MC, to produce the x^th plane is given by $M C(x)=M M_{0} x^{\log _{2} 0.9}$ , where $M_{0}$ = marginal cost to produce the first plane.[Note: You may use the fact that $\log _{2} x=\frac{\ln x}{\ln 2}$ .] If a plant produces planes with a 90% learning curve on production costs, and the marginal cost for the first plane is $700,000, then what is the marginal cost to produce the fourth plane? Round to the nearest thousand dollars.

You have a bank account that earns 9% nominal annual interest compounded continuously, and you want to have $90,000 in the bank account in five years so that you can buy a brand new Porsche.How much money would you have to deposit in one lump sum today so that the account balance would be $90,000 in five years? Round to the nearest dollar.

In March 1995 the space shuttle carried an experiment designed by a Harvard student who studies the growth of crystals.Suppose the probability density function of the length, x cm, of a crystal grown in space is modeled by $p(x)=x e^{-x}$ for $x \geq 0$ .The cumulative distribution function giving the probability that a crystal has length $\le$ t cm is represented by P(t).What is the mean crystal length that this model predicts? Give your answer to two decimal places.

Choose inequalities for r and $\theta$ which describe the following region in polar coordinates.

Set up an integral representing the volume of the torus (donut)obtained by rotating the circle $(x-6)^{2}+y^{2}=1$ about the y-axis.

A flat metal plate is in the shape determined by the area under the graph of $f(x)=\frac{1}{1+x}$ between x = 0 and x = 4.The density of the plate x units from the y-axis is given by x² grams/cm².Write down a Riemann sum which approximates the total mass.Convert it to an integral, and evaluate the integral to 3 decimal places.

Suppose that the distribution of family sizes in the city of Boston in the year 1956 was given by: Using this data, find the mean of family sizes in the city of Boston in the year 1956.Assume that all of the families with 7 or more members had precisely 7 members.Round to 2 decimal places.

The globular cluster M13 is a spherical distribution of stars which orbits our galaxy.Suppose that the density of stars in the cluster is purely a function of distance r from the center of the cluster and is given as $\rho(r)=\left(1+\left(\frac{r}{120}\right)^{3}\right)^{-5}$ stars/(ly)³, where r is measured in light-years, and 0 $\le$ r $\le$ 120 ly.(One light-year is the distance light travels in one year; "light-year" is abbreviated as "ly".)Set up an integral whose value is the exact number of stars in M13.

The force of gravitational attraction between a thin rod of mass M and length L and a particle of mass m lying on the same line as the rod at a distance of A from one of the ends is $\frac{G m M}{A(L+A)}$ .Use this result to select an integral for the total force due to gravity between two thin rods, both of mass M and length L, lying along the same line and separated by a distance A.[Hint: Divide one of the rods into small pieces, each of length dx and mass $\frac{M}{L} d x$ .Apply the formula above to each of the pieces, then form a Riemann sum.You know the rest…]

By the year 2016 you will have made your first million dollars.You invest it in a new company on January 1, 2017.The new company starts to earn a profit six months later.Thus, starting July 1, 2017, you receive income from the company in a continuous stream at a constant rate of 1/2 million dollars per year. Your bank offers interest at a nominal rate of 8% per year, compounded continuously.Consider the date when will you have received an income of $1 million from the company.(Do not take into account the bank's interest; this question is simply asking when the total income you have received will reach $1 million.)Which of the following is true?

Circle City is circular with a radius of 3 miles.Right in the center is a circular park with radius one mile.No one lives in the park.Elsewhere the population density is 4500(5 - r)people per square mile, where r is the distance from the center in miles.What is the average population density of Circle City, to the nearest whole number?

The probability density function f(x)shown below describes the chances that a computer circuit board will cost a manufacturer more than a certain number of dollars to produce.In this case, the cost of the circuit board, x, is measured in thousands of dollars.Which of the following definite integrals give the probability that the circuit card will cost between $2 thousand and some amount $b thousand? Assume that b is between 2 and 10.

The density of cars (in cars per mile)down a 24-mile stretch of the Massachusetts Turnpike starting at a toll plaza is given by $\rho(x)=500+100 \sin (\pi x)$ , where x is the distance in miles from the toll plaza and 0 $\le$ x $\le$ 24.Estimate the total number of cars down the 24-mile stretch by creating a Reimann sum, converting it to an integral, and evaluating it.