# Thread: [SOLVED] Lower and Upper Riemann sums?

1. ## [SOLVED] Lower and Upper Riemann sums?

Let $f(x) = sin x$. Estimate (integral from 0 to pi/2) $\int {f(x)} dx$ by calculating the lower Riemann sums $L(f,P_n)$ and the upper Riemann sums $U(f,P_n)$ for n = 3 with respect to a partition of $P_n$ $[0,\frac{\pi}{2}]$ into n subintervals of equal length. Verify that
$L(f,P_3)<= 1 <= U(f,P_3)$
and explain why $L(f,P_n) <= 1 <= U(f,P_n)$ for all $n > 0$

I don't even know what this means!!

2. Originally Posted by PTL
Let $f(x) = sin x$. Estimate (integral from 0 to pi/2) $\int {f(x)} dx$ by calculating the lower Riemann sums $L(f,P_n)$ and the upper Riemann sums $U(f,P_n)$ for n = 3 with respect to a partition of $P_n$ $[0,\frac{\pi}{2}]$ into n subintervals of equal length. Verify that
$L(f,P_3)<= 1 <= U(f,P_3)$
and explain why $L(f,P_n) <= 1 <= U(f,P_n)$ for all $n > 0$

I don't even know what this means!!
$n=3\Rightarrow\Delta{x}=\frac{\pi}{2n}=\frac{\pi}{ 6}$

$\text{ Let }x_i$ be the right end point of each subinterval, then

$x_i=(0+\frac{\pi{i}}{6})$, therefore

$U=\sum_{i=1}^{3}\sin{\frac{\pi{i}}{6}}*\frac{\pi}{ 6}$

Can you proceed?

3. Originally Posted by PTL
I don't even know what this means!!
I suggest you look for the definitions of an upper/lower Riemann sum in your notes or in your textbook. Once you know the definitions this problem will be easy to solve.

4. $P_{3} = \{ 0, \frac{\pi}{6} , \frac{\pi}{3}, \frac{\pi}{2} \}$

let $M_{i}$ be the supremum of f(x) on interval i, and let $m_{i}$ be the infinum of f(x) on interval i

then $M_{1} = \frac{1}{2}, \ M_{2} = \frac{\sqrt{3}}{2}, \ M_{3} = 1$

and $m_{1} = 0, \ m_{2} = \frac{1}{2}, \ m_{3} = \frac{\sqrt{3}}{2}$

and since the length of each interval is $\frac{\pi}{6}$

$U(f, P_{3}) = \frac {\pi}{6} \Big(\frac{1}{2}+\frac{\sqrt{3}}{2} +1\Big) \approx 1.24$

$L(f, P_{3}) = \frac {\pi}{6} \Big(0+\frac{1}{2} +\frac{\sqrt{3}}{2}\Big) \approx 0.715$

5. Here's a picture of the right sum

6. .

7. Thanks guys.

Question - how does one mark a thread as Solved?

Also - there's a bit at the end of the question, where I'm asked to
"Explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0".
Can I just say it's because the function is monotonically increasing and therefore the Upper Sum will be an overestimation, and the Lower Sum will be an underestimation?
Where does the '1' come into that though?

8. Originally Posted by PTL
Thanks guys.

Question - how does one mark a thread as Solved?

Also - there's a bit at the end of the question, where I'm asked to
"Explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0".
Can I just say it's because the function is monotonically increasing and therefore the Upper Sum will be an overestimation, and the Lower Sum will be an underestimation?
Where does the '1' come into that though?
The 1 is the actual area under the curve, i.e. $\int_0^{\pi/2} \sin x \, dx = 1$

9. Originally Posted by PTL
Question - how does one mark a thread as Solved?
Go to the top of the page, click on "Thread Tools" and then on "Mark this thread as solved".

10. Thanks.