Exam 8: Using the Definite Integral

A bridge worker needs to pull a 14-meter uniform cable with mass 3 kg/meter up to the work platform.How many joules are needed, assuming that the cable is hanging straight down from the platform? Round to the nearest whole number.

On each of two days (January 1, 2005 and January 1, 2006), a person deposits $2500 in a savings bank.The bank deposits interest in the account at a rate of 6%, compounded 4 times per year.Choose the sum that gives the size of the bank account at the end of 2006.

Consider the region bounded by the x-axis, the line x = 1, the line x = 4, and the curve $y=x^{2}$ .Imagine that this region is the base of a solid.Each cross-section of this solid perpendicular to the x-axis is a triangle of height H . The volume of this solid can be found exactly and can be written as $\frac{H B}{6}$ .What is B?

Let p(t)be a probability density which is defined for 0 $\le$ t $\le$ 1.Could the following be the cumulative distribution function for p? (Remember that the cumulative distribution function at time t is the integral of p from 0 to t.)

Find the arc length of $r=2 \sin \theta$ , $0 \leq \theta \leq \pi$ .Give an exact answer and then round to 2 decimal places.

A straight road goes through the center of a circular city of radius 5 km.The density of the population at a distance r (in km)from the road is well approximated by $D(r)=30-4 r$ (in thousand people per km²).Find the total population of the city, to the nearest person.

A 3-gram drop of thick red paint is added to a large can of white paint.A red disk forms and spreads outward, growing lighter at the edges.Since the amount of red paint stays constant through time, the density of the red paint in the disk must vary with time.Suppose that its density p in gm/cm² is of the form $p=k(t) f(r)$ for some functions k(t)of time and f(r)of the distance to the center of the disk.Let R(t)be the radius of the disk at time t.Which of the following equations expresses the fact that there are 3 grams of red paint in the disk?

An object is in the shape drawn below; its boundary is obtained by rotating the parabola $y=2 x^{2}$ (for 0 $\le$ x $\le$ 1)around the y axis.(Units are in centimeters.)Suppose that the density of this object varies with height according to the rule $\rho$ (y)= 12 (2 - y)grams/cm³.Which of the following Riemann sums computes (approximately)the mass in grams of this object?

Suppose that the distribution of people's ages in the United States is essentially constant, or uniform, from age 0 to age 60, and from there it decreases linearly until age 100.This distribution p(x)is shown below, where x is age in years, and p measures probability density.Such a probability distribution is called trapezoidal. According to this simplified model of the distribution of people's ages in the United States, what percentage of the population is between 0 and 100 years old?

In a recent archaeological expedition, a scroll was discovered containing a description of a plan to build what appears to be the Tower of Babel.According to the manuscript, the tower was supposed to have a circular cross section and "go up to the heavens" (i.e., be infinitely high).A mathematician was consulted to solve some of the questions posed by the archaeologists.The mathematicians plotted half of the silhouette of the tower on a set of coordinate axes with the y-axis running through the center and discovered that it was approximated by the curve $y=-100 \ln \left(\frac{x}{6}\right)$ .Would such a tower have finite volume? If so, calculate it to the nearest whole number.If not, enter "infinite".

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.What is the probability that the circuit board will cost less than $2 thousand to produce?

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 6% per year, compounded continuously.Suppose T is measured in years from January 1, 2017.Compute the future value of your original investment of $1 million at time T and the future value at time T of the income that you have received by that time.Use these to determine how many years, T, it will take for your investment to have paid off.Round to 2 decimal places.

Consider the 3-sided region bounded by $x=0$ , $y=2$ and $y=\sqrt{x}$ .What is the perimeter of this region? Give your answer to 3 decimal places.

Somebody offers to pay you money in one of the following ways: • Two $50 payments, one six months from now and one twelve months from now. • Payment in a continuous cash flow over the next year at a constant rate of $99 per year. These payments are to be deposited into a bank account that earns 10% interest compounded continuously.You want to determine which plan is preferable, i.e., which plan has a larger present value.Find the present value of the first payment plan.Then use a Riemann sum converted into a definite integral to get the exact present value of the second payment plan.Which payment plan has the larger present value?

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

The price of crude oil in the past can be approximated by $P(t)=40-(t-4)^{2}$ , where P(t)is measured in $US/barrel, and time t is measured in months, with t = 0 on July 1, 1990.In the same time period, Saudi Arabia produced oil at a rate approximated by $R(t)=160+30 \arctan (t-3)$ (measured in million barrels per month).Assume that the oil is sold continuously two months after its production.How many billions of dollars did Saudi Arabia get for the oil it produced in the last 6 months of 1990? Round to 1 decimal place.

On each of two days (January 1, 2005 and January 1, 2006), a person deposits $500 in a savings bank.The bank deposits interest in the account at a rate of 8%, compounded continuously.Choose the sum/integral that gives the size of the bank account at the end of 2006.

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 following formulas gives the cumulative distribution function P(t)for t $\ge$ 0?

A cone has base radius = 4 cm and height = 2 cm.Using horizontal slicing, which of the following definite integrals represents volume of the cone?

A curve is given parametrically as $x=4 \sin t$ and $y=4 \cos t$ for 0 $\le$ t $\le$ 2 $\pi$ .How does the length change if the coefficients of x and y are doubled (meaning x = 8 sin t and y = 8 cos t)?