Question

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places...

Answer 1

integral from a to b
(b-a)/n = base length
a+(base length/2) = x value of first height
+ base length = x value of next height
+ base length = etc
the first time is (base length / 2) because we want midpoint between a and a+baselength
then the points following would simply be base length distance away

this is just the steps... (because you didn't give the integral the question is asking about x'D)

Answer 2

O_O

WAIT NO WHAT
I WASN'T DONE TYPING AAAA

Answer 3

lol, I mean, you posted it like half an hour ago and just didn't finish x'D

Answer 4

Bleh I'm sorry x_x

Answer 5

OKAY FIRST ONE

Answer 6

\[\Large{\lim_{n\rightarrow\infty}\int_{0}^{\pi}e^{x}cos{x}dx,~n=4}\]

Answer 7

I thought we did this already?

Answer 8

*checking closed Q records*

Answer 9

OH DARN IT ALL

Answer 10

I clearly remembering having to draw out pi/8 stuff for you

Answer 11

I'm sorry :'c

Answer 12

lol np x'D
do you still need help with it?

Answer 13

well I got as far as this:\[\large{\sum_{i=1}^{n=4}(e^{x_{i}})\cos{x_{i}}\Delta x}\rightarrow(e^{x_{i}})\cos{x_{i}}\left(\frac{\pi-0}{4}\right)\]\[\Large{\rightarrow(e^{x_i})\cos{x_i}\left(\frac{\pi}{4}\right)}\]

Answer 14

we're uhhh using the midpoint rule right?

Answer 15

Yeah

Answer 16

ohhhh yeah, you're doing it right
nvm x'D

Answer 17

the sigma sign should still be in front though, I think

Answer 18

So all \(\large{x_1}\) are multiples of \(\Large{\frac{\pi}{8}}\) because interval \(\large{\frac{\pi}{4}\div4=\frac{\pi}{8}}\)

Answer 19

professor said take it off the minute I put in a numerical value idk

Answer 20

first x = pi/8
then next point is x = pi/8 + pi/4 = 3pi/8

Answer 21

yup so
I put this as my sum:\[[~f(\pi/8)+f(3\pi/8)+f(5\pi8)+f(7\pi/8)~](\pi/8)\]

Answer 22

perfect! :)

Answer 23

Okay that is good lol

Answer 24

wait

Answer 25

?

Answer 26

base length is pi/4

Answer 27

!!
I FORGOT TO CHANGE THAT OFF MY PAPER
*really irritated with self*

Ok, all good otherwise?

Answer 28

ye

Answer 29

☺YAY

Answer 30

Okay so... next one is\[\large{\int_{1}^{3}\frac{2}{1+x^2}dx,~n=5}\]

Answer 31

Riemann form\[\large{\sum_{i=1}^{5}\frac{2}{1+x_i^2}\Delta x}\rightarrow\frac{2}{1+x_i^2}\left(\frac{3-1}{5}\right)\rightarrow\frac{2}{1+x_i^2}\left(\frac{2}{5}\right)\]

Answer 32

yup

Answer 33

Midpoint formula = \(\large{\frac{\Delta x}{2}=\frac{2/5}{2}=5}\)?

Answer 34

WAIT NO\[\frac{2/5}{2}=\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}\]yeah?

Answer 35

yeah
so first x = 1 + 1/5 = 6/5

Answer 36

ah yeah

Answer 37

then second x = 6/5 + 2/5 = 8/5

Answer 38

so basically...\[x_1=\frac{6}{5}\times\frac{2}{5}i\]

Answer 39

oops that is supposed to be a plus sign

Answer 40

o-o
I gues??

Answer 41

oh yeah
then that's right x'D

Answer 42

okay that's it for this question... if I have more I'll open up a new one so everyone gets medals fairly 😂