Question

use your choice of a) simpsons rule b) the trapezoidal rule or c) the midpoint rule to approximate pi using the equation
pi= integral (4/(1+x^2)) from 0 to 1

Answer 1

what was a?

Answer 2

missing info

Answer 3

which rule do you like best?

Answer 4

that's all the info

Answer 5

simpsons is a quadratic fit

trap is an average rectangle fit

midpoint is a rectangle fit

Answer 6

midpoint rule i guess..whichever one is easier

Answer 7

midpoint is simplest, is this spose to be limits, or are you spose to do an approximation?

Answer 8

what choice di you use of a

Answer 9

did*

Answer 10

to approximate, we would want to define a subinterval length

Answer 11

it says experiment with the number of subintervals until you obtain successive approximations that differ by less than 10^(-3)

Answer 12

a is a simpson rule choice

Answer 13

ah, then we will want to do this a few times :)

Answer 14

since the interval is of length 1, then any division of it into subintervals is /n for n subintervals, agreed?

Answer 15

*is 1/n for n ......

Answer 16

so the sum of height times width defines the area of a rectangle

the width is 1/n, the height is [i(1/n) + (i+1)(1/n)]/2 for some leftside i

if that makes any sense

Answer 17

hmmm, thats the trap on second thought

Answer 18

yes agreed

Answer 19

so i + half the subinterval is the midpoint between i and i+1

so i+1/(2n)

spose our subinterval is a length of 1/10

the midpoint to use for x in our function for the intercal from 7/10 to 8/10 is 7/10 + 1/20

Answer 20

once we know the areas, we add them up

Answer 21

f(x) = 4/(1+x^2)

lets try a subinterval of length 1/10

f(0/10+1/20) * 1/10
+f(1/10+1/20) * 1/10
+f(2/10+1/20) * 1/10
+f(3/10+1/20) * 1/10
...
+f(9/10+1/20) * 1/10
-------------------
approx area of integration

Answer 22

i spose

i/10 + (i+1)/10
--------------
2

is the midpoint/average between 2 x values with in a length of 1/10

2i + 1
------
20

Answer 23

if you have excel, you can create the values quite easily