Exam 5: Itegration

Given a consumer's demand function, $D ( q ) = \frac { 300 } { 0.4 q + 8 }$ dollars per unit,find the total amount of money consumers are willing to spend to get 15 units of the commodity.

An object is moving so that its velocity after t minutes is $v ( t ) = 1 + 8 t + 9 t ^ { 2 }$ meters per minute.How far does the object travel from the end of minute 2 to the end of minute 3?

Given an initial population, $P _ { 0 } = 140,000$ ,a renewal rate,R = 355,and a survival function, $S ( t ) = e ^ { - 000 t }$ ,with time t measured in years,determine the population at the end of 19 years.

Evaluate $\int \left( 5 x ^ { 6 } - 9 x + 7 \right) d x$ .

A certain nuclear power plant produces nuclear waste in the form of strontium-90 at the constant rate of 200 pounds per year.The waste decays exponentially with a half-life of 28 years.How much of the radioactive waste from the nuclear plant will be present after 130 years? [Hint: Think of this as a survival and renewal problem.] Round to two decimal places.

Money is transferred continuously into an account at the constant rate of $1,100 per year.The account earns interest at the annual rate of 8% compounded continuously.How much will be in the account at the end of 3 years?

Evaluate $\int 5 x ^ { 4 } e ^ { x ^ { 5 } } d x$ .

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve $y = \sqrt { 49 - x ^ { 2 } }$ from x = -7 to x = 7.

Evaluate $\int \left( \frac { x } { 9 } - 9 \right) ^ { 2 / 9 } d x$ .

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 11 } d x$ .

Find the area of the region under the line y = 4x + 7 over the interval 1 $\leq$ x $\leq$ 8 .

In a certain section of the country,the price of chicken is currently $3 per kilogram.It is estimated that x weeks from now,the price will be increasing at a rate of $3 \sqrt { x + 1 }$ cents per kilogram.How much will chicken cost 4 weeks from now?

The velocity v(t)(t)= x'(t)at time t of an object moving along the x axis is given,along with the initial position x(0)of the object. $x ^ { t } ( t ) = - 3 ( 3 t + 1 ) ^ { 1 / 2 }$ ; x(0)= 4 Find: (a)The position x(t)at time t. (b)The position of the object at time t = 1. (c)The time when the object is at x = 2. Round answers for parts (b)and (c)to one decimal place.

The slope f '(x)at each point (x,y)on a curve y = f (x)is given,along with a particular point (a,b)on the curve.Use this information to find f (x). $f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }$ ; (-6,1)

Find the Gini index for the given Lorentz curve.Round your answer to four decimal places,if necessary. $L ( x ) = 0.6 x ^ { 2 } + 0.3 x$

Water flows into a tank at the rate of $\sqrt { 2 t + 7 }$ ft³/min.If the tank is empty when t = 0,how much water does it contain 8 minutes later? Express the answer to two decimal places.

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day.By how much will the value of the crop increase during the next 10 days if the market price remains fixed at $6 per bushel?

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 4 } d x$ .

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 3 + 3 x ) x ^ { - 1 / 2 }$ people per month.By how much will the population increase over the next 9 months?

Evaluate the indicated integral: $\int \frac { 7 x ^ { 4 } + 56 x ^ { 3 } + 56 } { x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 } d x$