Question

can someone please thoroughly explain each step to this calc problem pleasE??

Answer 1

Are you familiar with Reimann Sums?

Answer 2

yes kinda, i dont have my textbook with me thats why im so stuck

Answer 3

\[\int_a^b f(x)dx = \sum_{i=1}^n f(x_i)dx \]

Answer 4

Please refer to http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

Answer 5

i just put up an example so someone could explain it step by step and the work so i can apply it to my homework questions(:

Answer 6

Start by finding your \(\Delta x\)\[\Delta x = \frac{b-a}{n}\]This will tell you where your right endpoint starts.

Answer 7

And the change in your x-value to your next endpoint.

Answer 8

what is b and a ?

Answer 9

Your equation: \(\int_{-2}^2 (x^3+8)dx\)

Compare to: \(\int_a^b f(x)dx\)

Answer 10

1

Answer 11

So: \[\Delta x = \frac{b-a}{n} \implies \frac{2-(-2)}{4} = 1\]

Answer 12

Yep. so your first rectangle starts at x=1.

Answer 13

Since \(\Delta x = 1\), your second rectangle is at x=2

Answer 14

Ok, once you find all your right endpoints, you will use the formula \[R_4 = \sum_{i=1}^4 f(x_i)\Delta x\]

Answer 15

That means
\[\begin{align}
R_4 = \sum_{i=1}^4 f(x_i)\Delta x &=
\\&= f(x_1)\Delta x +f(x_2)\Delta x +f(x_3)\Delta x + f(x_4)\Delta x
\end{align}\]

Answer 16

so first rectangle is at 1,second is at 2, third is at 3, and fourth is at 4?

Answer 17

Yep.

Answer 18

Only because the change is \(\Delta x=1\)

Answer 19

oh this isnt that bad, i get it so far(:

Answer 20

Yay!

Answer 21

Glad you're comprehending what I am explaining to you.

Answer 22

so then for each \(f(x_i)\) you will pluf the \(x\)-value into your original equation: \(x^3+8\) to find the exact value, then multiply it by \(\Delta x\).

Answer 23

then add them all up and you're done! :)

Answer 24

wait i got kinda confused on the last step, can you show me an example?

Answer 25

or do it with me please??

Answer 26

\[x=1~,~2~,~3~,~4\]so find : \(f(1)~,~f(2)~,~f(3)~,~f(4)\)

Answer 27

Yes you are right :)

Answer 28

isnt there two questions? the height and to approximate?

Answer 29

No, the total area is 131, the height of each rectangle is just the value you get when you plug in your x value.

Answer 30

So technically we answered two questions for the price of one :P

Answer 31

Yes :D

Answer 32

mmhmm

Answer 33

ok just making sure, thanks(:

Answer 34

And you know why it's an approximate? Because you're only finding the area w.r.t the right endpoints.

I'm sure you learned in class that to find the exact area you have to subtract the overestimate by the underestimate?

Answer 35

yesss

Answer 36

That's why :)

Answer 37

thank you so much, you explained it very well

Answer 38

Thank you!