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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.
:::

(1)
<iframe src="https://www.desmos.com/calculator/c64xzaskbk?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Value of the Riemann Sum: 10.7820774995
:::info
(2) Evaluate $M_3$ using Desmos.
:::

(2)
<iframe src="https://www.desmos.com/calculator/xppt3axdi5?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Value of the Riemann Sum: 14.8990552326

:::info
(3) Evaluate $L_9$ using Desmos.
:::

(3)
<iframe src="https://www.desmos.com/calculator/k6ed00egxy?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Value of the Riemann Sum: 18.8231531881
:::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/nyd7ys7xsn?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Value of the Riemann Sum: 15.7677319241
:::info
(5) Evaluate $R_{1000}$ using Desmos.
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(5)
<iframe src="https://www.desmos.com/calculator/7ddfgdhaw8?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

Value of the Riemann Sum: 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 $x=10$.
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(6)
$$\int_{a}^{b} f(x) dx$$
$$\int_{1}^{10}\left|\frac{10x}{x^2+1}\sin \left(x\right)\right|+\frac{4}{x^2+1} dx$$

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