Math 182 Miniproject 2 Numerical Methods of Integration.md
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Math 182 Miniproject 2 Numerical Methods of Integration
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**Overview:** In this project we find exact formulas for integral approximations using Riemann sums of various flavors.

**Prerequisites:** Section 5.6 of _Active Calculus_ and a strong background in $\sum$-notation.

We have learned multiple ways of approximating the values of integrals.
* $R_n$ --- Right Riemann sum using $n$ rectangles
* $L_n$ --- Left Riemann sum using $n$ rectangles
* $T_n$ --- Trapezoid Riemann sum using $n$ rectangles
* $M_n$ --- Midpoint Riemann sum using $n$ rectangles
* $S_{2n}$ --- Simpson's rule using $n$ intervals

Evaluate each of the following. Let Desmos crunch the numbers on each sum for you. Just be sure to include the expressions that you used to set up the calculation.

__Problem 1.__
$\int_4^{10}\sin(x)\,dx$

$$\int_{ }^{ }\sin\left(x\right)dx=cos(x)+C$$

__Evaluate the integral with the specified bounds and use Desmos to calculate:__

$$[cos(10)]-[cos(4)]=0.185427908213$$

__Problem 2.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $R_{100}$.

__Given that the interval from 4 to 10 is 6 units long and there are 100 right-handed rectangles:__

$$dx=6/100=0.06$$

$$R_{100}=\sum_{i=4}^{n\ =\ 100}(1+0.06(i))\cdot 0.06=0.191755718035$$

__Problem 3.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $L_{100}$.

__Given that there are 100 left-handed rectangles:__

$$L_{100}=\sum_{i=3}^{n\ =\ 100}(1+0.06(i))\cdot 0.06=0.17898883497$$

__Problem 4.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $T_{100}$.

__Since $T_{100}$ is the average of $R_{100}$ and $L_{100}$, use the following formula to find $T_{100}$:__

$$T_{100}=\frac{R_{100}+L_{100}}{2}$$

$$T_{100}=\frac{(0.191755718035)+(0.17898883497)}{2}=0.185372276502$$

__Problem 5.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $M_{100}$.

$$M_{100}=\sum_{i=4}^{n\ =\ 100}f(1+(\frac{0.06)(i)}{2})\cdot 0.06=0.18545572532$$

__Problem 6.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $S_{200}$.

__Since $S_{200}$ is the weighted average of $M_{100}$ and $T_{100}$, using the formula for Simpson's Rule:__

$$S_{200}=\frac{2(M_{100})+T_{100}}{3}$$

$$S_{200}=\frac{2(0.18545572532)+(0.185372276502)}{3}=0.185427909047$$

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