Thread: Upper and Lower Limits of interval for Riemann's Sum

1. 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!

2. Re: Upper and Lower Limits of interval for Riemann's Sum

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 $\int_0^3 {1dx}$

The upper limit is $\int_0^3 {(1+x)dx}$

3. Re: Upper and Lower Limits of interval for Riemann's Sum

ohhh so: lower = 3, upper = 3+3x?

4. Re: Upper and Lower Limits of interval for Riemann's Sum

Originally Posted by simba412
ohhh so: lower = 3, upper = 3+3x?
not quite ...

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

as Plato stated ...

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

the lower limit is 3, but the upper limit is not 3 + 3x

5. 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)???

6. Re: Upper and Lower Limits of interval for Riemann's Sum

oh wait... (1 + (3-0))3 = 12?

...

8. 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