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)
Using the Desmos Riemann Sum Application to calculate the right-hand Riemann sum for$f(x)$:

$R_3=\sum_{i=1}^{n\ =\ 3}(1+3(i))\cdot 3=10.7820774995$


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(2) Evaluate $M_3$ using Desmos.
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(2)
Using the Desmos Riemann Sum Application to calculate the midpoint Riemann sum for$f(x)$:

$M_3=\sum_{i=1}^{n\ =\ 3}f(\bar{x_i})\cdot 3=14.8990552326$


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(3) Evaluate $L_9$ using Desmos.
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(3)
Using the Desmos Riemann Sum Application to calculate the left-hand Riemann sum for$f(x)$:

$L_3=\sum_{i=0}^{n\ =\ 3-1}(1+3(i))\cdot 3=27.6694261007$


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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)
As calculated for the right-hand Riemann Sum $R_{100}$:

$R_{100}=\sum_{i=1}^{n\ =\ 100}(1+0.09(i))\cdot 0.09=15.7677319241$

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(5) Evaluate $R_{1000}$ using Desmos.
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(5)
As calculated for the right-hand Riemann Sum $R_{1,000}$:

$R_{1,000}=\sum_{i=1}^{n\ =\ 1,000}(1+0.009(i))\cdot 0.009=15.9945370554$

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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)
$\lim_{n \to \infty}\sum_{i=1}^{n=10 }(|\frac{10(x_{i})}{(x_{i})^2+1}sin(x_{i})|+\frac{4}{(x_{i})^2+1})\cdot \triangle(x)$

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