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

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.*

---

:::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.
:::

(1)
$\Delta x=\frac{10-1}{3}=3$ units wide
$R_{3}= 3*f(4)+3*f(7)+3*f(10)$
$R_{3}= 3(2.016+1+.578)$
$R_{3}= 10.782$
:::info
(2) Evaluate $M_3$ using Desmos.
:::

(2)
$\Delta x=\frac{10-1}{3}=3$ units wide
$M_{3}= 3*f(2.5)+3*f(5.5)+3*f(8.5)$
$M_{3}= 3(2.615+1.37+.981)$
$M_{3}= 14.898$
:::info
:::info
(3) Evaluate $L_9$ using Desmos.
:::

(3)
$\Delta x=\frac{10-1}{9}=1$ units wide
$L_{9}= 1(f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7)+f(8))$
$L_{9}= 1(6.207+4.437+.823+2.016+1.998+.561+1+1.279+.501)$
$L_{9}= 18.822$
:::info
(4) Evaluate $R_{100}$ using Desmos. You will probably want to use the $\sum$-notation capabilities of Desmos.
:::

(4)
<iframe src="https://www.desmos.com/calculator/jdtyrc1xqg?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Based on the results found from the notation capabilities of Desmos, we know from the graph that $R_{100}=15.7677$
:::info
(5) Evaluate $R_{1000}$ using Desmos.
:::

(5)
<iframe src="https://www.desmos.com/calculator/qzzgr283eh?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>
The graph above shows the Riemann Sums notation capabilities of Desmos, which shows $R_{1000}=15.9945$
:::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 $x=10$.
:::

(6)
a=1 b=10 $\Delta x=\frac{10-1}{n}=\frac{9}{n}$

Area=$\lim_{n \rightarrow \infty}\sum_{i=1}^{n\ \ }[|\frac{10(1+\frac{9i}{n})}{(1+\frac{9i}{n})^2+1}sin(1+\frac{9i}{n})|+\frac{4}{(1+\frac{9i}{n})^2+1}]\frac{9}{n}$

---

To submit this assignment click on the Publish button . Then copy the url of the final document and submit it in Canvas.