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\centerline {\largerbold Problems, Week 5}

\bigskip

%improper integrals: problems 1 through 4
%trig integral: problem 5
%diff equations (sort of!) problems 6 & 7



1.~Sketch the curves $y=e^{-2x}$ and $y=e^{-3x}$ carefully. Is the area
enclosed between them in the first quadrant finite?  

\bigskip

2.~Consider the following four integrals:
\medskip
\settabs 2 \columns
\+a) $\displaystyle{ \int_0^{\infty} {x \over {1+x^4}} \, dx}$
& b) $\displaystyle{ \int_0^{\infty} {{x^2} \over {1+x^4}} \, dx}$\cr
\medskip
\+c) $\displaystyle{ \int_0^{\infty} {{x^3} \over {1+x^4}} \, dx}$
&d) $\displaystyle{ \int_0^{\infty} {{x^4} \over {1+x^4}} \, dx}$\cr
\medskip
Which of these integrals converge? (Hint: compare to the ``correct'' pure 
powers of $x$.) Compute the exact value of at least 
one of the convergent integrals.

\bigskip

3.~a) The following improper integral converges:
$\displaystyle{\int_0^{\infty} x e^{-x^2} \, dx}$. What is its value?

\medskip

b) The value of $\d{\int_0^{\infty} e^{-x^2} \, dx}$ (another convergent
improper integral) is $\displaystyle{{\sqrt \pi} \over 2}$ (this amazing
fact easiest to explain with some of the tools in third semester
calculus). Improper integrals involving polynomials and $e^{-x^2}$ often
arise in statistics (and hence in statistical analyses of
experiments). Use integration by parts to get a formula relating
$\d{\int_0^{\infty} x^n e^{-x^2} \, dx}$ and $\d{\int_0^{\infty} x^{n-2}
e^{-x^2} \, dx}$, where $n$ is a positive integer bigger than 2. (The
parts you take will be slightly tricky.)

\medskip

c) Now find the values of
\settabs 3 \columns
\+i) $\displaystyle{\int_0^{\infty} x^2 e^{-x^2} \, dx}$ 
&ii) $\displaystyle{\int_0^{\infty} x^3 e^{-x^2} \, dx}$ 
&iii) $\displaystyle{\int_0^{\infty} x^4 e^{-x^2} \, dx}$\cr
\smallskip 
You will need the reduction formula and the two initial values given in a) 
and b).

\bigskip

4.~The integral $\d {\int_1^{\infty} {{e^{- 5x}} \over {1+x^2}} \, dx}$
certainly converges. One simple way to try to approximate its value is to
write it as the sum of two integrals:

$$\int_1^{\infty} {{e^{-5x}} \over
{1+x^2}} \, dx = 
\int_1^{A} {{e^{-5x}} \over
{1+x^2}} \, dx + 
\int_A^{\infty} {{e^{-5x}} \over
{1+x^2}} \, dx$$

where $A$ is some large number. Then any of our numerical techniques
could be used to approximate the first (proper!) definite integral on the
right-hand side of the equation. We could just neglect the second
integral if we knew that it was small.

a) Compare the second integral on the right to an integral over the same
interval of a simpler function to try to find an $A$ so that the improper
integral to be dropped will have a value smaller than $10^{-5}$.

\medskip 

b) Use a numerical technique to try to evaluate the first integral with
an error less than $\pm 10^{-5}$.  {\it You will definitely need a
programmable calculator or a computer for this part!}  \medskip c)
Compute $\d {\int_1^{\infty} {{e^{-5x} \over {1+x^2}} \, dx}}$ with an
error less than $\pm 10^{-4}$.  

Note: there is {\bf no} unique or best possible answer to this question,
either for the ``simpler'' function or for the $A$.

\bigskip

5.~a) If $m$ and $n$ are constants, compute $\d{\int_0^{2\pi} \cos mx
\; \cos nx \; dx}$.

\medskip

b) Suppose $f(x)= A \cos x + B \cos 2x + C \cos 3x$, and that you also
know

$$\int_0^{2\pi}  f(x) \cos x \; dx = 5 \qquad
\int_0^{2\pi}  f(x) \cos 2x \; dx = 6 \qquad
\int_0^{2\pi}  f(x) \cos 3x \; dx = 7\; .$$

Find $A$ and $B$ and $C$.

\smallskip Note: this is a computational beginning to Fourier series,
a way of analyzing periodic phenomena.

\bigskip

6.~If a function $F$ is continuous on an interval $[a,b)$ and if
$\d{\lim_{x \to b^-} F(x) = +\infty}$ then we'll say ``$F$ 
explodes at $b$.''

\smallskip

a) At what point do the functions $\d{{{x^2+1}\over {x-1}}}$ and
$\d{{{\cos x} \over {x-2}}}$ explode?

b) Write all solutions to the differential equation $y'=y^2$ subject to
the initial condition $y(0)=y_0$. Can you find a solution that explodes
at 10? And another one that explodes at 5? Can you find a solution that
explodes at $x_0$ when $x_0 > 0$? How does the initial condition at $x=0$
connect with a specified explosion at $x_0$? Draw a picture of an
exploding solution.

\bigskip

7.~A cylindrical metal tank, of height 10 feet, has circular cross section
of radius 2 feet.  There is a small hole in the bottom of the tank, through
which water leaks at a rate proportional to the depth of the water in the
tank.  The tank is filled at noon and at 2:00 the depth of water in the
tank is recorded to be 3 feet.
\medskip
(a) Find the volume of water in the tank $t$ hours after noon.
\medskip
(b) Suppose that the water leaking from this tank is caught in a second
tank in the shape of a right circular cone, whose top has radius 2 feet and
whose height is 10 feet.  At what rate is the depth of water in this tank
increasing an instant before the tank begins to overflow?
\bigskip

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