# Upper and Lower Limits of interval for Riemann's Sum

• Nov 18th 2012, 02:26 PM
simba412
Upper and Lower Limits of interval for Riemann's Sum
So i tried to this question and got f(x) = 1 + x
upper estimate = 9
lower estimate = 6

wondering if i'm on the right path. thx!
• Nov 18th 2012, 02:38 PM
Plato
Re: Upper and Lower Limits of interval for Riemann's Sum
Quote:

Originally Posted by simba412
So i tried to this question and got f(x) = 1 + x
upper estimate = 9
lower estimate = 6

No.

The lower limit is $\displaystyle \int_0^3 {1dx}$

The upper limit is $\displaystyle \int_0^3 {(1+x)dx}$
• Nov 18th 2012, 02:45 PM
simba412
Re: Upper and Lower Limits of interval for Riemann's Sum
ohhh so: lower = 3, upper = 3+3x?
• Nov 18th 2012, 02:53 PM
skeeter
Re: Upper and Lower Limits of interval for Riemann's Sum
Quote:

Originally Posted by simba412
ohhh so: lower = 3, upper = 3+3x?

not quite ...

$\displaystyle 1 \le f(x) \le 1+x$

as Plato stated ...

$\displaystyle \int_0^3 dx \le \int_0^3 f(x) \, dx \le \int_0^3 1+x \, dx$

the lower limit is 3, but the upper limit is not 3 + 3x
• Nov 18th 2012, 03:09 PM
simba412
Re: Upper and Lower Limits of interval for Riemann's Sum
i thought the upper limit was (1+x)dx where dx = 3, therefore 3(1+x)???
• Nov 18th 2012, 03:13 PM
simba412
Re: Upper and Lower Limits of interval for Riemann's Sum
oh wait... (1 + (3-0))3 = 12?
• Nov 18th 2012, 03:33 PM
skeeter
Re: Upper and Lower Limits of interval for Riemann's Sum
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• Nov 18th 2012, 03:46 PM
simba412
Re: Upper and Lower Limits of interval for Riemann's Sum
thanks for the graph. i think i got it! the lower limit represents, in this case, the area of the red rectange. the upper limit represents the the area of the triangle + rectange multiplied by 2 since we take the upper value of x therefore = 4x3 = 12