Question

hi

Answer 1

@Kainui

Answer 2

@sleepyjess @johnweldon1993 @Preetha

Answer 3

Oh, I totally need to learn this!
This is the one thing my teacher didn't teach us in Calculus this year.

Answer 4

+ the AP test is this upcoming week zzz.

Answer 5

http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx

Answer 6

im not in calc yet. sorry.

Answer 7

Try reading the article I sent you and check if you can make any sense of it, because I certainly can't right now.

Answer 8

https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/defintsoldirectory/DefIntSol2.html#SOLUTION 10
Refer to this possibly?

Answer 9

Solution #10

Answer 10

Sampling point in your problem would be ck = 2k/n

Answer 11

I don't understand where the (2/3) came from in that explanation.

Answer 12

Oookaay, this is kinda hard to explain :P

Answer 13

So i'm going to try and take a intuitive explanation rather than a rigorous way to prove everything...

Answer 14

Hmmm, if you want to find out properly then look up the "definition of riemann integral"

Answer 15

I know the Riemann integral, can you explain where the 2/3 came from?

Answer 16

is it the delta x?

Answer 17

The definition of a definite integral is:
\[
\int_a^b f(x) \ dx = \lim_{n\to\infty} \sum_{k=1}^n f(x_k) \Delta x = \lim_{n\to\infty} \sum_{k=1}^n f(x_k) \frac{b - a}{n}
\]

Try to identify the parts of your problem with the definition.

Answer 18

\[\int\limits_{0}^{2}\], but can you assume that? Could it possible be 4 over 2?

Answer 19

@andrewyates
f(xk) = 5 + 2k/n

Answer 20

Plug in k = 1 to find the first x coordinate and then add 2 to find the bounds... I think

Answer 21

what would n be though?

Answer 22

n is the number of rectangles you're using to approximate the area under the curve. As n approaches infinity, the number of rectangles become infinite and the width of each approaches 0 so the approximation becomes exact. Here's an image: https://upload.wikimedia.org/wikipedia/commons/2/2a/Riemann_sum_convergence.png

Answer 23

@Hero
please help savior

Answer 24

\[
\int_5^7(x+5)^{10} \ dx
\]

Answer 25

Okay, I believe that's the answer.

Answer 26

Hey, I had the (x+5)^10 right from before xD

Answer 27

Nice

Answer 28

Until I deleted it, :P

Answer 29

Want me to explain? @shubham1

Answer 30

Can you quickly explain how you got the limits as 7 and 5?

Answer 31

I know they have to have a delta value of 2.

Answer 32

Actually, the bounds are from 0 to 2. My mistake.

Answer 33

\( k\frac{2}{n} \) represents the x value in the sum, so plugging in k = 1 (first x value) gives us \(\frac{2}{n}\) and as \(n\to\infty\) it becomes 0.

Answer 34

Add 2, and we get the bounds \(x \in [0, 2]\)

Answer 35

Oh wow, I had the whole equation from the start without even knowing it, haha lright thanks!

Answer 36

Okay, cool!