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) The right hand Riemann Sum using 3 boxes is approximately equals 10.7820 units.

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(2) Evaluate $M_3$ using Desmos.
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(2) The middle Riemann Sum using 3 boxes is approximatey equals 14.8990 units.
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(3) Evaluate $L_9$ using Desmos.
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(3) The left hand Riemann Sum using 9 boxes is approximatey equals 18.8231 units.

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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) The right hand Riemann Sum using 100 boxes is approximately equals 15.7677 units.

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(5) Evaluate $R_{1000}$ using Desmos.
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(5) The right hand Riemann Sum using 1000 boxes is approximately equals 15.9945 units.

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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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$\int_1$$^1$$^0$$$\left|\frac{10x}{x^2+1}\sin \left(x\right)\right|+\frac{4}{x^2+1}
$$

The function is supposed to be closer to the intergral but I do not know how to do that.

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