Question

CHECK MY ANSWER!!!!
Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x^3 and y = x.

Answer 1

x^3=x
x= -1, 0, 1
y=0, -1, 1

deltax= (b-a)/n
deltax= (1-(-1)/4
deltax= 2/4 = 0.5

[-1, 0]
midpoint= 1/2 (-1+0)
midpoint= 1/2(-1) = -0.5

[0, 1]
midpoint= 1/2 (1+0)
midpoint= 1/2(1) = 0.5

area[-1,1] = 0.5 (0.5+(-0.5))
area[-1,1] = 0.5 (0)
= 0

Answer 2

@Evoker

Answer 3

I'm looking, trying to follow your work, looks like you need f(-1),f(-.5),f(0),f(.5), and f(1)

Answer 4

I assume for midpoint rule you want the sum of (delta x)(f(mid upper)-f(mid lower))

Answer 5

Yea

Answer 6

Quickly looking up the method are we using f(-1), f(-.5) or f(-.75) for the first region.

Answer 7

Ah midpoints so yes the first needs to be f(-.75)

Answer 8

so -.75 for one and -.421875 for the other which yields a difference of .328125 and multiplied time .5 yields an area for the first region of .1640625

Answer 9

yes, right so its f(-0.75), f(-0.5), f(-0.25) then f(0.25), f(0.50), f(0.75)

Answer 10

well four regions so f(-.75),f(-.25),f(.25),f(.75)

Answer 11

ok

Answer 12

Should be able to repeat what I did for the first and then add all the region together.

Answer 13

Wont it still be zero

Answer 14

[-1, 0]
f(-0.75) + f(-0.25)


[0, 1]
f(0.75) + f(0.25)

area[-1,1] = f(-0.75) + f(-0.25) + f(0.75) + f(0.25)
area[-1,1] = -0.421875 + -0.0625 + 0.421875 + 0.0625
= 0

Answer 15

Are you doing negative area because if not you want to make the first two positive

Answer 16

Geometrically they all should have positive results

Answer 17

True, I'm not quite sure what they are looking for

Answer 18

I think you should make them all positive and then add them up or in essence twice the sum of the last two.

Answer 19

ok so
positive area:
area[-1,1] = f(-0.75) + f(-0.25) + f(0.75) + f(0.25)
area[-1,1] = 0.421875 + 0.0625 + 0.421875 + 0.0625
= 0.96875

Answer 20

Yes that would be my answer

Answer 21

ok I have more questions stick around!!!!

Answer 22

well don't forget the .5 base for each

Answer 23

oh yea

Answer 24

0.96875 (0.5)

Answer 25

hmm by the way I got, .328125 for the first region before the .5 element

Answer 26

how?

Answer 27

Ok once again for the first region you need to take the difference of f(-.75) for the two functions.

Answer 28

for y=x this gives a result of -.75 while y=x^3 yields -.428175

Answer 29

The difference is .328125

Answer 30

Oops right

Answer 31

So repeating for the second region f(-.25) gives -.25 and -.015625 yielding a difference of .234375

Answer 32

Third region f(.25) gives .25 and .015625 yielding the same difference of .234375

Answer 33

And fourth will be the same as first

Answer 34

ok one minute

Answer 35

positive area:
area[-1,1] = deltax (f(-0.75) + f(-0.25) + f(0.75) + f(0.25))
area[-1,1] = 0.5 ((0.75-0.421875) + (0.25-0.0625) + (0.75-0.421875) + (0.25-0.0625))
area[-1,1] = 0.5 ((0.328125) + (0.1875) + (0.328125) + (0.1875))
= 0.515626

Answer 36

@Evoker

Answer 37

I had instead of .1875 a result of .234375 for f(.25) and f(-.25)

Answer 38

.015625 for the cubed term not .0625

Answer 39

yea, I squared it

Answer 40

positive area:
area[-1,1] = deltax (f(-0.75) + f(-0.25) + f(0.75) + f(0.25))
area[-1,1] = 0.5 ((0.75-0.421875) + (0.25-.015625) + (0.75-0.421875) + (0.25-.015625))
area[-1,1] = 0.5 ((0.328125) + (0.234375 ) + (0.328125) + (0.234375 ))
= 0.5625

Answer 41

@Evoker

Answer 42

Looks good now

Answer 43

Great. Next question!

Answer 44

Ok