Question

The expression (1/50)(sqrt(1/50)+sqrt(2/50)+sqrt(3/50)+...+sqrt(50/50) is a Riemann Sum approximation for
They want you to find its integral equivalent

Answer 1

Well set up its integral equivalent, anyways

Answer 2

It was multiple choice, but my teacher says that we need to know how to do all of them as if we weren't given the answer choices

Answer 3

is there no n's in your sum thingy above?

Answer 4

Nope. They want you to set up a definite integral with it

Answer 5

I know what the answer is, but I have no idea how to get to it

Answer 6

does this work ?\[\int\limits_0^1 \sqrt{x}~dx\]

Answer 7

oh approximation

Answer 8

Yes, thats exactly it

Answer 9

the given riemann sum looks like "Upper sum with 50 intervals"

Answer 10

How did you do that

Answer 11

Whoooa

Answer 12

\[\frac{b-a}{n}=\Delta x \\ \text{ choosing } b=1 \text{ and } a=0 \text{ we have } \frac{1}{n} \\ \frac{1}{n} \cdot \sum_{i=1}^n \sqrt{0+i \cdot \frac{1}{n}} \\ \lim_{n \rightarrow \infty} \sum_{i=1}^n \sqrt{\frac{i}{n}} \frac{1}{n} =? \]

Answer 13

Is n the number of subintervals?

Answer 14

yeah

Answer 15

What is i?

Answer 16

it is good to memorize that
\[\lim_{n \rightarrow \infty}\sum_{i=1}^{n} f(x_i) \Delta x=\int\limits_{a}^{b}f(x) dx \\ \text{ where } x_i=a+i \cdot \Delta x \\ \text{ and } \Delta x =\frac{b-a}{n}\]

Answer 17

There is no way that I will be able to memorize that

Answer 18

\[\sum_{i=1}^{n}f(x_i) \Delta x\\ \Delta x(f(x_1)+f(x_2)+f(x_3) \cdots +f(x_n))\]

Answer 19

oh my god

Answer 20

it is just a whole bunch of areas for rectangles being adding

Answer 21

i guess i can see how it looks way more scarier than that

Answer 22

Why is it just sqrt x inside of the integral?

Answer 23

replace f(xi) with f(x)
your xi is a+i delta x
your xi is 0+i*1/n in this case

Answer 24

oh ok

Answer 25

wait...no...

Answer 26

actualy yayayaya sorry :D

Answer 27

\[\int\limits_{a}^{b}f(x) dx=\lim_{n \rightarrow \infty} \sum_{i=1}^n f(x_i) \Delta x \\ \text{ where} \Delta x=\frac{b-a}{n} \text{ and } x_i=a+i \Delta x \\ \int\limits_a^b f(x) dx= \lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(a+i \Delta x) \Delta x \\ \text{ Example } \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(5+i \frac{10}{n}) \cdot \frac{10}{n} \\ \frac{b-a}{n}=\frac{10}{n} \\ \text{ and we know } a=5 \text{ so we can find } b \\ \frac{b-5}{n}=\frac{10}{n} \\b=15 \\ \text{ so we have } \int\limits_5^{15} f(x) dx=\lim_{n \rightarrow \infty}\sum_{i=1}^{n}f(5+i \frac{10}{n}) \cdot \frac{10}{n}\]

Answer 28

so if f was sin(x)
you would put sin(x)

Answer 29

or if f was kitty(x) you would put kitty(x)

Answer 30

http://gyazo.com/57bc0b924b5b9df6d6c99c55a597ea94

Answer 31

\[\lim_{n \rightarrow \infty}\sum_{i=1}^{n}kitty(5+i \cdot \frac{10}{n}) \cdot \frac{10}{n}=\int\limits_{5}^{15} kitty(x) dx\]

Answer 32

where kitty is a mind boggling function of x

Answer 33

Thank you!

Answer 34

here is a more nice animation

Answer 35

lol did u make that

Answer 36

yes haha just testing what all geogebra can do :P

Answer 37

haha looks like u are having fun