Question

Compute approximate value of definite integral using trapezoid rule and simpsons rule? Please help?

Answer 1

Question is in the file attachment

Answer 2

do you know what the trapezoid rule is?

\[\int\limits_{a}^{b} e^{x^{2}} \approx (b-a)\left[ \frac{ f(a) + f(b)}{ 2} \right]\]

Answer 3

simpsons rule is

\[\int\limits_{a}^{b}f(x)dx= \frac{ b-a }{ 6 }\left[ f(a)+4f(\frac{ a+b }{ 2 })+f(b) \right]\]

Answer 4

I thought that was the trapezoid rule

Answer 5

you showed the trapezium rule

Answer 6

O okay so how do I do this one then?

Answer 7

oh it is the trapezoid rule, it was the same as i said lol one sec i gotta re write the working out

Answer 8

b-a=2-0=2
b-a=2

we have n=4 so we want to divide the whole thing by 4
since we are dividing by 4, we divide b-a by 4 to get the height of each strip
this gives 2/4=0.5


so we will do values 0, 0.5, 1, 1.5, 2

now we have the formula

\[(2-0)\left[ \frac{ f(0)+f(0.5)+f(1)+f(1.5)+f(2) }{ 4 } \right]\]

Answer 9

Wait what about the simpson

Answer 10

just sub into the formula i gave you for simpsons rule

Answer 11

pretty straight forward

Answer 12

wait so was that for the trapezoid rule? and was the first formula that u gave me wrong?

Answer 13

nah we had the same formula

Answer 14

Okay Gimme a sec

Answer 15

I will give u ur best response points but lemme see how it turns out for the simpsons rule so one second

Answer 16

okay it is right i confused myself let me know if you need anything explained

Answer 17

question.

Answer 18

for the first part. why did you add 0, 0.5, 1, 1.5, and 2

Answer 19

f(0) + f(0.5) + f(1) + f(1.5) +f(2)

Answer 20

we have n=4 so we want to find the intervals to which we use as our values x0 x1 x2 x3 x4 x5 etc
since n=4 we divide the difference of a and b by n
this gave us 0.5
so the intervals of 0.5 gaves us the values
x0=0
x1=0.5
x2=1
x3=1.5
x4=2

Answer 21

then we did f(x0)+f(x1)+...f(xn)

Answer 22

okay so you did that for just 0-2 right

Answer 23

say it was f 4-0 then i would put 8 values right?

Answer 24

yes because our limits only go from 0 to 2

Answer 25

0-4**

Answer 26

got it

Answer 27

here is your best response points. but i still haven't done this for the simpsons rule so i will ask you if i have any problems. thx arnie

Answer 28

if you had limits from 0 to 4
and n=4
you would do 4-0 then divide by 4
this gives intervals of 1
therefore you get
x0=0
x1=1
x2=2
x3=3
x4=4

Answer 29

wait why is the denominator 4?

Answer 30

isn't it supposed to be 2?

Answer 31

it will simplify to 1/2