DesignProject$-2-$In this design project, you will write a program that calculates the $$area under the curve$$ of

a polynomial function. The user provides the polynomial$$s coefficients, and the program

returns the $($approximate$)$ value of the area.

#include

#include

#include

#include

#include

using namespace std;

const double XMIN $=-2\mathrm{.}0$;

const double XMAX $=2\mathrm{.}0$;

const double TOL $=0\mathrm{.}00001$;

$//$ pretty$-$print polynomial

void print$\_$polynomial$($vector c$)\{$

double ci;

bool flag $=$ true;

cout $<<"$p$($x$)="$;

for$($unsigned long int i $=$ c$.$size$()-1$; i $>=0$; i$--)\{$

ci $=$ c$.$at$($i$)$; $//$ current coefficient

if $($c$.$size$()==1)$ cout $<<$ ci; $//$ always print at least one coefficient

if $($ci $!=0)\{$

$//$ Print signs. Only print first sign if it is negative.

if $($flag$)\{$

$//$ handle first minus sign

flag $=$ false;

if $($ci $<0)$ cout $<<"-"$;

$\}$ else $\{$

if $($ci $<0)$ cout $<<"-"$;

else cout $<<"+"$;

$\}$

$//$ Print coefficient

if $($i $==0)$

cout $<<$ abs$($ci$)$; $//$ always print last coefficient

else if $($abs$($ci$)!=1)$

cout $<<$ abs$($ci$)$; $//$ don't print ones

$//$ Print x$,$ except for last coefficient

if $($i $!=0)$

cout $<<"$x$"$;

$//$ Print exponent, except for last two coefficients

if $($i $>1)$

cout $<<"^"<<$ i;

$\}$

$\}$

cout $<<$ endl;

$\}$

$//$ evaluate polynomial given by coefficients c at x

double eval$($double x$,$ vector c$)\{$

$//$ your code here

$\}$

$//$ exact $($theoretical$)$ area

double exact$($vector c$)\{$

vector c$\_$int; $//$ coefficients of integral

c$\_$int.push$\_$back$(0\mathrm{.}0)$; $//$ integral of polynomial has no zero$-$degree term

for $($unsigned long int i $=0$; i $<$ c$.$size$()$; i$++)\{$

c$\_$int.push$\_$back$($c$.$at$($i$)/($i$+1))$; $//$ integrate term$-$by$-$term

$\}$

return eval$($XMAX$,$ c$\_$int$)-$ eval$($XMIN$,$ c$\_$int$)$; $//$ fundamental theorem of calculus

$\}$

$//$ read coefficients from cin, until empty newline

vector read$\_$coefficients$($void$)\{$

$//$ your code here

$\}$

double estimate$($double step, vector c$)\{$

$//$ your code here

$\}$

int main$($void$)\{$

$//$ your code here

return $0$;

$\}$

ie if the Input is:

$-4$

$-4$

$10$

$-1$

the output will be

Solution:

p$($x$)=-$x$^3+10$x$^2-4$x $-4$

Estimate $=37\mathrm{.}33334707$

Exact $=37\mathrm{.}33333333$

Error $=0\mathrm{.}00001373$

Iterations $=20$

Step $=0\mathrm{.}00000038$