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title: 'Math 181 Miniproject 11: Riemann Sums'

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Math 181 Miniproject 11: Riemann Sums.md
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tags: MATH 181
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Math 181 Miniproject 11: Riemann Sums
===

**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 $x=10$.

![](https://i.imgur.com/h56UdIm.png)


Before you start, enter the function $f(x)$ into Desmos so that you can refer to it later.
(1) Evaluate $R_3$ using Desmos.
:::    

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

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

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

   
:::info
(4) Evaluate $R_{100}$ using Desmos. You will probably want to use the $\sum$-notation capabilities of 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:
$f(x_k)Δx$

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

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

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

$R_100=15.7677319241$
    
:::info
(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:
$f(x_k)Δx$

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

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

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

$R_1000=15.9945370554$
    
:::info
(6) Write out an expression using a limit that will give the exact area under the curve $y=f(x)$ from $x=1$ to $x=10$.
:::

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

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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