# Use a double integral to find the volume of the solid shown

uestion

1.) Use a double integral to find the volume of the solid shown in

the figure.

? cu units

2.) Use the method of Lagrange multipliers to minimize the

function subject to the given constraints.

Minimize the function f(x, y) = xy subject to the constraints x2 +

4y2 = 4 and x ≥ 0.

Minimum of ___ at (x , y) = (

)

3.) f is a joint probability density function for the random

variables X and Y on D. Find the indicated probability.

f(x, y) = xy; D = {(x, y) | 0 ≤ x ≤ 1; 0 ≤ y ≤ 2}

P(x + 2y ≤ 1) = ?

4.) "The Fountains of Bellagio" is a choreographed water display

set to light and music that takes place in front of the Bellagio

Hotel in Las Vegas. In the evening, the shows take place every

15 min from 7 P.M. to midnight. The duration of each show is 7

min. If Joan arrives at a random time between 8 P.M. and 8:30

P.M. for an evening show, find the probability that Joan will have

to wait the following.

(a) less than 7 min before the next show begins

(b) more than 8 min before the next show begins

5.) Suppose that the time intervals in seconds between arrivals

of successive cars at an expressway tollbooth during rush hour

are exponentially distributed with associated probability density

function f(t) = (1/5)e−(1/5)t.

Find the probability that the average time interval between

arrivals of successive cars is more than 5 sec. (Round your

answer to three decimal places.)

6.) The rate at which the concentration of a drug in the

bloodstream decreases is proportional to the concentration at

any time t. Initially, the concentration of the drug in the

bloodstream is C0 g/mL. What is the concentration of the drug in

the bloodstream any time t? Formulate but do not solve the

problem in terms of a differential equation with a side condition.

(Let C(t) denote the concentration at any time t and k (positive)

be the constant of proportion.)

dC

=

dt

C(0) =

7.) Suppose a quantity Q(t) does not exceed some number C; that

is,Q(t) ≤ C for all t. Suppose further that the rate of growth of

Q(t) is jointly proportional to its current size and the difference

between C and the natural logarithm of its current size. What is

the size of the quantity Q(t) at any time t? Formulate but do not

solve the problem in terms of a differential equation with a side

condition. The graph of Q(t) is called the Gompertz growth curve.

This model, like the ones leading to the learning curve and the

logistic curve, describes restricted growth. (Let Q0 denote the

size of the quantity present initially and k (positive) be the

constant of proportion.)

dQ/dt =

Q(0) =

8.) Use a double integral to find the volume of the solid shown in

the figure.

_____ cu units

9.) Solve the first-order differential equation by separating

variables. (Use C for the constant of integration. Remember to

use ln(|u|) where appropriate.)

y’ = (17y ln x) / x

10.) Find the solution of the initial value problem. (Remember to

use ln(|u|) where appropriate.)

y’ = 5 − y; y(0) = 4

11.) Assume that the rate of change of the supply of a

commodity is proportional to the difference between the demand

and the supply, so that

dS/dt = k(D – S)

where k is a constant of proportionality. Suppose that D is

constant and S(0) = S0. Find a formula for S(t).

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