Riemann Sums

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September 3rd 2010, 05:08 PM
stobs2000

Riemann Sums

hi this is a Riemann sum q:

Find U, the Riemann Upper Sum for f(x) = x2 on [0,2], using 4 equal sub-intervals

i know that n=0.5 so i thought the upper limit would be 0.5^2+ 1^2+1.5^2+2^2 which gives 7.5 but the answer is 3.75 so could someone please explain why this is the answer?

cheers
September 3rd 2010, 05:16 PM
stobs2000

i realised i forgot to multiply the whole thing by n! so i figured out that question but can someone help me with this one:

When calculating the Riemann Upper and Lower Sums (U and L) for the function f(x) = x2 on the interval [0,2], what is the smallest number of (equal) sub-intervals needed to make U - L ≤ 0.1 ?
September 3rd 2010, 06:26 PM
skeeter

Quote:

Originally Posted by stobs2000

i realised i forgot to multiply the whole thing by n! so i figured out that question but can someone help me with this one:

When calculating the Riemann Upper and Lower Sums (U and L) for the function f(x) = x2 on the interval [0,2], what is the smallest number of (equal) sub-intervals needed to make U - L ≤ 0.1 ?

I assume $f(x) = x^2$
(fyi, use the caret to signify exponents ... x^2)

$\displaystyle \frac{2}{n}\left[f(x_1) + f(x_2) + ... + f(x_{n})\right] - \frac{2}{n}\left[f(x_0) + f(x_1) + ... + f(x_{n-1})\right] \le 0.1$

finish it.
September 3rd 2010, 06:42 PM
stobs2000

thanks-according to that i get 80 which is the answer but i don't understand where you got 2/n from
September 3rd 2010, 07:26 PM
stobs2000

Here is another question:

Given that $cos\sqrt{x}$ decreases on the interval [0,9], estimate the value of $\int_{0}^{9}cos\sqrt{x} dx$ using the Riemann Lower Sum L on this interval with three unequal sub-intervals [0,1], [1,4], [4,9]. Enter your answer correct to two decimal places.

I know that n for the first 3 terms is 1/3, for the next 3 it is 1 and the next 3 it is 5/3 so to find the lower riemann sum i did:

$(cos\sqrt{0}+cos\sqrt{1/3}+cos\sqrt{2/3})$ x1/3 $+(cos\sqrt{1}+cos\sqrt{2}+cos\sqrt{3})$ x1 $+(cos\sqrt{14/3}+cos\sqrt{19/3}+cos\sqrt{8})$ x 5/3

and if i use radians mode on the calculator that gives me -2.49

ps- sorry about the above- im just learning hoe to use latex editor
September 4th 2010, 04:41 AM
skeeter

Quote:

Originally Posted by stobs2000

thanks-according to that i get 80 which is the answer but i don't understand where you got 2/n from

how would you define the width of each subinterval?
September 4th 2010, 05:09 AM
skeeter

Quote:

Originally Posted by stobs2000

Here is another question:

Given that $cos\sqrt{x}$ decreases on the interval [0,9], estimate the value of $\int_{0}^{9}cos\sqrt{x} dx$ using the Riemann Lower Sum L on this interval with three unequal sub-intervals [0,1], [1,4], [4,9]. Enter your answer correct to two decimal places.

I have no idea what you are doing in this calculation. There are three subintervals given, so you do not need to calculate n. The first subinterval has a width of 1, the second 3, and the third 5.

Right Riemann sum using the given subintervals ...

$\cos{{\sqrt{1}} \cdot 1 + \cos{{\sqrt{4}} \cdot 3 + \cos{{\sqrt{9}} \cdot 5 \approx -5.66$

btw ... next time, start a new problem w/ a new thread.
September 4th 2010, 03:14 PM
stobs2000

thanks X1000