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) $R_3$ is ~10.78

$R_3$~$(f(4)*3)+(f(7)*3)+(f(10)*3)$

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(2) Evaluate $M_3$ using Desmos.
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(2) $M_3$ is ~ 14.90

$M_3$~$(f(2.5)*3)+(f(5.5)*3)+(f(8.5)*3)$

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(3) Evaluate $L_9$ using Desmos.
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(3) $L_9$ is ~ 18.82

$\sum_{i=1}^{9}[f(1+k*(1))*(1)$

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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) $R_{100}$ is ~ 15.77

$\sum_{i=1}^{100}\left[f(1+k\cdot\frac{9}{100}\right)]\cdot\frac{9}{100}$

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(5) Evaluate $R_{1000}$ using Desmos.
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(5) $R_{1000}$ is ~ 15.99

$\sum_{i=1}^{1000}\left[f(1+k\cdot\frac{9}{1000}\right)]\cdot\frac{9}{1000}$

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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)

A= $lim_{n-> ∞}$ $\sum_{i=1}^{n}\left[f(1+k\cdot\frac{9}{n}\right)]\cdot\frac{9}{n}$

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