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

August 30, 2017

Question
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
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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