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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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) To begin this process what I did was was evaluate R3 using Desmos like the question asked. Then,using the R3 Riemann Right sums equation and plug in the numbers that are given like:
n=3 Δx= upper - lower/ n
then the points that come out of the formula would be: 4, 7, and 10.
We can find f(4), f(7), and f(10) by plugging in said numbers into our original equation:
f(7)=$\frac{10\left(7\right)\sin\left(7\right)}{7^{2}+1}+\frac{4}{7^{2}+1}$ to get 0.999. Doing the same thing with the other points, f(4)=2.016 and f(10)=0.578. With these values we can use the equation below to solve by pluggingin the values:
$\sum_{1}^{3}f\left(I\right)=Δx=3\left[f\left(4\right)+f\left(7\right)+f\left(10\right)\right]$ then doing all the math that equals 10.779 for R3

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(2) Evaluate $M_3$ using Desmos.
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(2) For this part we are trying to find the middle points for the M3 that is the middle points:
$\sum_{1}^{3}f\left(I\right)=Δx$
that comes out to be the points are 2.5 , 5.5 , 8.5
all of those using the formula from the last question, those answers would be f(2.5) = 2.615 , f(5.5) = 1.369 and lastly for f(8.5)= 0.981 then combine all the numbers above we get a grand total of 14.895 for the final answer for $M3$.

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(3) Evaluate $L_9$ using Desmos.
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(3) For this part this is trying to find now the lower parts of the problem and we use the Riemann sums equation once again so:
n=9 or Δx = $\frac{\left(10-1\right)}{9}$ which equals 1
All of the points for this would be 1,2,3,4,5,6,7,8,9, and 10. The left over points would be [1,2,3,4,5,6,7,8,and 9]
So, doing all the math in the end it would be $\sum_{1}^{9}f\left(I\right)=Δx$ equaling 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) Now to find what we are looking for with R100 we would do:
n=100 and a=1
then following the rules of the formula it would equal I=16.27
this is a different equation it is:
$I=\sum_{i=0}^{n=1}f\left(s\left(I\right)\right)\cdot w$ and that is how we got our answer for this part.

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(5) Evaluate $R_{1000}$ using Desmos.
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(5) so now again we use that same formula:
$I=\sum_{i=0}^{n=1}f\left(s\left(I\right)\right)\cdot w$
to now find R1000 and when all calculated out the final answer for R1000 is 16.045.

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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) For this part to find the exact curve we want to use the formual following:
$\sum_{1}^{10}f\left(i\right)\ Δx$
$Δx=\frac{x_{u}x_{l}}{n}\ and\ n\ equaling\ \infty$
then following all the steps it is:
A: $\int_{x=1}^{10}f\left(x\right)\ dx$
then solving everything out the final answer becomes:
$A=\int_{x=1}^{10}\left[\frac{10x}{x^{2}+1}\sin\left(x\right)+\frac{4}{x^{2}+1}dx\right]$.

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