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)
$f(1)=\frac{10(1)}{1^2+1}sin(1)+\frac{4}{1^2+1}=6.207$
$f(2)=\frac{10(2)}{2^2+1}sin(2)+\frac{4}{2^2+1}=4.437$
$f(3)=\frac{10(3)}{3^2+1}sin(3)+\frac{4}{3^2+1}=0.823$
$f(4)=\frac{10(4)}{4^2+1}sin(4)+\frac{4}{4^2+1}=2.016$
$f(5)=\frac{10(5)}{5^2+1}sin(5)+\frac{4}{5^2+1}=1.998$
$f(6)=\frac{10(6)}{6^2+1}sin(6)+\frac{4}{6^2+1}=0.561$
$f(7)=\frac{10(7)}{7^2+1}sin(7)+\frac{4}{7^2+1}=0.999$
$f(8)=\frac{10(8)}{8^2+1}sin(8)+\frac{4}{8^2+1}=1.279$
$f(9)=\frac{10(9)}{9^2+1}sin(9)+\frac{4}{9^2+1}=0.501$
$f(10)=\frac{10(10)}{10^2+1}sin(10)+\frac{4}{10^2+1}=0.578$
$R_3$ sum for n=3 $\triangle{}x=xupper-\frac{xlower}{n}$
or
$\triangle{}x=\frac{10-1}{3}$ $\triangle x=3$
The points are {1,4,7,10}
{4,7,10} are the correct points and $R_3$ is given by
$R_3=\sum^3_1 f(i)\triangle x=3[f(4)+f(7)+f(10)]$
Which equals 10.779

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(2) Evaluate $M_3$ using Desmos.
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(2) $M_3$ sum is n=3 and $\triangle x=3$
The points are 1,4,7,10
The middle points are {2.5,5.5,8.5}
$M_3=\sum^3_1f(i)\triangle x$
$f(2.5)=\frac{10(2.5)}{2.5^2+1}sin(2.5)+\frac{4}{2.5^2+1}=2.615$
$f(5.5)=\frac{10(5.5)}{5.5^2+1}sin(5.5)+\frac{4}{5.5^2+1}=1.369$
$f(8.5)=\frac{10(8.5)}{8.5^2+1}sin(8.5)+\frac{4}{8.5^2+1}=1.981$
$M_3=3(2.615+1.369+0.981)$
$=14.895$

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(3) Evaluate $L_9$ using Desmos.
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(3) $L_9$ is sum n=9 $\triangle x=\frac{10-1}{9}=1$
Points are 1,2,3,4,5,6,7,8,9,10
Left points are {1,2,3,4,5,6,7,8,9}
$L_9=\sum^9_1f(i)\triangle x=19.399$
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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)
n=100, a=1
$I=\sum^{^n-1}_{n=0}if(S(i))*w$
$I=16.27435$

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(5) Evaluate $R_{1000}$ using Desmos.
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(5)
$n=1000, a=1$
$I=\sum^{n-1}_{n=0}f(s(i))*w$
$I=16.0452$

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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)
$\sum^{10}_{1}f(i)\triangle x$ and $\triangle x=\frac{X_u-X_l}{n}n->infinity$
$A=\int_{x=1}^{10}f\left(x\right)dx$
$A=\int_{x=1}^{10}\left[\frac{10x}{x^{2}+1}\sin\left(x\right)+\frac{4}{x^{2}+1}\right]dx$

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