Solution of ${\displaystyle \underset{-1}{\overset{1}{}}}\sqrt[\phantom{2}]{1-x^2}dx$ vsing PRV$.$

$1$ #question in the lecture note:

$2$ import numpy as np

$3$

$4n=1000000$

$5$ success $=0$

$6$ #we make a for loop to creat a random numbers and count the success points

$7$

$8$ for $i$ in range $(n)$ :

$9$ #make random point $(,yi)$

$10,=np.$random.uni form $(-1,1)$ #the range of $x$ is from $-1$ to $1($one side of the box$)$

$11yi=np.$random.uniform $(,1)$ #the range of $x$ is from $$ to $1($The other side of the box$)$

$12,fi=$ np$.$sqrt $(1-x{}^{**}2),||f()$ function at $$

$13$ #check if $yi$ is less than $f()$

$14$ if yi $$ fxi:

$15$ success $=$ success $+1$

$16$

$17$ rateOfSuccess $=$ success $\frac{?}{n}$ #success points over the total number of points

$18$ areaOfRectangle $=2,$#$1^{*}2=2$ is the area of the box ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=$ green aren $-$ red area $0$

$19$ integral $=$ rateOfSuccess$*areaOfRectangle$ #formula for estimation $of$ area under the curve

$20$

$21$ print$(integral)$

$1\mathrm{.}57091$

$ofhe$ function has negative valre $we$ should break the interval $to$ two piece

one with negative value ome with possible value and make a box for each one.

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=$ green aren $-$ red area

$if$ the furetion $is$ above xaais

$we$ jist need $to$ modity

the highlighted part $of$ the code.$-1$ and $0$

$19$ integral $=$ rateOfSuccess$*areaOfRectangle$ #formula for estimation $of$ area under the curve

$20$

$21$ print$(integral)$

$1\mathrm{.}57091$

$ofhe$ function has negative valre $we$ should break the interval $to$ two piece

one with negative value ome with possible value and make a box for each one.

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=$ green aren $-$ red area

$if$ the furetion $is$ above xaais

$we$ jist need $to$ modity

the highlighted part $of$ the code.$($Total of $6$ marks$)$ Use a random number generator in Python $($e$.$g$.,$ numpy.random.uniform$())$ to ap$-$ import numpy as np

$3$

$4$n $=1000000$

$5$ success $=0$

$6$ #we make a for loop to creat a random numbers and count the success points

$7$

$8$ for i in range$($n$)$:

$9$ #make random point $($xi$,$yi$)$

xi $=$ np$.$random.uniform$(-1,1)$ #the range of x is from $-1$ to $1($one side of the box$)$

yi $=$ np$.$random.uniform $(0,1)$ #the range of x is from $0$ to $1($The other side of the box$)$

fxi $=$ np$.$sqrt$(1-$xi$**2)$ #f$($xi$)$ function at xi

#check if yi is less than f$($xi$)$

if yi $$ fxi:

success $=$ success $+1$

rateOfSuccess $=$ success$/$n #success points over the total number of points

areaOfRectangle $=2$ #$1*2=2$ is the area of the box $-1$

proximate the following integral:

${\displaystyle \underset{-2}{\overset{2}{}}}3x^2-6xdx.$

Show your code with the result.

Hint: The function is positive on $-2,0,$ and negative on $0,2.$ You should use the method taught in

class with at least $100,000$ points.

Use method shown in image