Math 181 Miniproject 11: Riemann Sums.md
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tags: MATH 181
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Math 181 Miniproject 11: Riemann Sums
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**Overview:** This miniproject focuses on the use of $\sum$-notation to estimate the area under a curve. Students will use Desmos to set up and evaluate Riemann sums to get the area under a curve that is not amenable to the Fundamental Theorem of Calculus.

**Prerequisites:** Section 4.3 of *Active Calculus.*

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:::info
For this miniproject you will be estimating the area under the curve
$$
f\left(x\right)=\left|\frac{10x}{x^2+1}\sin \left(x\right)\right|+\frac{4}{x^2+1}
$$
from $x=1$ to $x=10$.


Before you start, enter the function $f(x)$ into Desmos so that you can refer to it later.
(1) Evaluate $R_3$ using Desmos.
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(1) $f\left(1\right)=\frac{10\left(1\right)}{1^{2}+1}\sin\left(1\right)+\frac{4}{1^{2}+1}=6.207$
$f\left(2\right)=\frac{10\left(2\right)}{2^{2}+1}\sin\left(2\right)+\frac{4}{2^{2}+1}=4.437$
$f\left(3\right)=\frac{10\left(3\right)}{3^{2}+1}\sin\left(3\right)+\frac{4}{3^{2}+1}=0.823$
$f\left(4\right)=\frac{10\left(4\right)\sin\left(4\right)}{4^{2}+1}+\frac{4}{4^{2}+1}=2.016$
$f\left(5\right)=\frac{10\left(5\right)\sin\left(5\right)}{5^{2}+1}+\frac{4}{5^{2}+1}=1.998$
$f\left(6\right)=\frac{10\left(6\right)\sin\left(6\right)}{6^{2}+1}+\frac{4}{6^{2}+1}=0.561$
$f\left(7\right)=\frac{10\left(7\right)\sin\left(7\right)}{7^{2}+1}+\frac{4}{7^{2}+1}=0.999$
$f\left(8\right)=\frac{10\left(8\right)\sin\left(8\right)}{8^{2}+1}+\frac{4}{8^{2}+1}=1.279$
$f\left(9\right)=\frac{10\left(9\right)\sin\left(9\right)}{9^{2}+1}+\frac{4}{9^{2}+1}=0.501$
$f\left(10\right)=\frac{10\left(10\right)\sin\left(10\right)}{10^{2}+1}+\frac{4}{10^{2}+1}=0.578$
So that being said R3 Riemann Right sum for n=3 Δx=xupper - xlower/n OR Δx=$\frac{10-1}{3}$ $Δx=3$
then the points would be 1,4,7,10 and {4,7,10} are the right points
and R3 is given by:
R3= $\sum_{1}^{3}$ f(i) Δx = 3 $\left[f\left(4\right)+f\left(7\right)+f\left(10\right)\right]$
then this would equal 3(3.583) and that = 10.779

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(2) Evaluate $M_3$ using Desmos.
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(2) M3 or midde Riemann sum for n = 3 and Δx = 3
those points are 1,4,7,10 and {2.5,5.5,8.5} are the middle points
then M3= $\sum_{1}^{3}$ f(i) Δx
f(2.5)= $\frac{10\left(2.5\right)\sin\left(2.5\right)}{2.5^{2}+1}+\frac{4}{2.5^{2}+1}=2.615$
$f\left(5.5\right)=\frac{10\left(5.5\right)\sin\left(5.5\right)}{5.5^{2}+1}+\frac{4}{5.5^{2}+1}=1.369$
$f\left(8.5\right)=\frac{10\left(8.5\right)\sin\left(8.5\right)}{8.5^{2}+1}+\frac{4}{8.5^{2}+1}=0.981$
Then M3 = 3(2.615+1.369+0.981) = 3(4.965) = 14.895

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(3) Evaluate $L_9$ using Desmos.
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(3) L9 or Left Riemann Sum for n=9 or Δx= $\frac{10-1}{9}=1$
then those points are 1,2,3,4,5,6,7,8,9,10 and {1,2,3,4,5,6,7,8,9} are those left points
then L9= $\sum_{1}^{9}$ f(i) Δx and that equals 19.399

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(4) Evaluate $R_{100}$ using Desmos. You will probably want to use the $\sum$-notation capabilities of Desmos.
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(4) n=100, a=1
$I=\sum_{i=0}^{n-1}f\left(s\left(i\right)\right)\cdot w$
So that being said and beigng able to plug everything in then that would equal $I=16.2743523821$

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(5) Evaluate $R_{1000}$ using Desmos.
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(5) n= 1000 a=1
$I=\sum_{i=0}^{n-1}f\left(s\left(i\right)\right)\cdot w$
So that being said and beigng able to plug everything in then that would equal
$I=16.0451991012$

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(6) Write out an expression using a limit that will give the exact area under the curve $y=f(x)$ from $x=1$ to $x=10$.
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(6) The exact area under curve = $\sum_{1}^{10}$ f(i) Δx
and $Δx=\frac{X_{u}-X_{l}}{n}$ $n→\infty$
so:
A=$\int_{x=1}^{10}$ f(x) dx
A= $\int_{x=1}^{10}\left[\frac{10x}{x^{2}+1}\sin\left(x\right)+\frac{4}{x^{2}+1}\right]dx$

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