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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$Δx= \frac{b-a}{n}$
$Δx= \frac{10-1}{3}$
$Δx= 3$
$3(f(4)+f(7)+f(10))$
$𝑅_3=10.7820774995$

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(2) Evaluate $M_3$ using Desmos.
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$Δx= 3$
$3(f(2.5)+f(3.5)+f(8.5))$
$M_3=14.4752543066$

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(3) Evaluate $L_9$ using Desmos.
:::
$Δx= 3$
$3(f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7)+f(8)+f(9))$
$L_4=56.4694595643$

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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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$Δx= \frac{9}{n}$
$x_1=1+Δx$
$x_2=1+2Δx$
$x_3=1+3Δx$
$x_k=1+2kΔx$

Thus, the area of the Kth rectangle is:
$f(x_k)Δx$

$=f(1+k\frac{9}{n})\frac{9}{n}$

$\sum_{k=1}^{n}f(1+k\frac{9}{n})\frac{9}{n}$

$\sum_{n=1}^{100}f(1+n(0.09))(0.09)$

$R_100=15.7677319241$

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(5) Evaluate $R_{1000}$ using Desmos.
:::

$Δx= \frac{9}{n}$
$x_1=1+Δx$
$x_2=1+2Δx$
$x_3=1+3Δx$
$x_k=1+2kΔx$

Thus, the area of the Kth rectangle is:
$f(x_k)Δx$

$=f(1+k\frac{9}{n})\frac{9}{n}$

$\sum_{k=1}^{n}f(1+k\frac{9}{n})\frac{9}{n}$

$\sum_{n=1}^{1000}f(1+n(0.009))(0.009)$

$R_1000=15.9945370554$

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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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$Δx= \frac{9}{n}$
$x_1=1+Δx$
$x_2=1+2Δx$
$x_3=1+3Δx$
$x_k=1+2kΔx$

Thus, the area of the Kth rectangle is:
$f(x_k)Δx$

$=f(1+k\frac{9}{n})\frac{9}{n}$

$\sum_{k=1}^{n}f(1+k\frac{9}{n})\frac{9}{n}$

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