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title: Math 182 Miniproject 2 Numerical Methods of Integration

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

$\int_4^{10}\sin(x)\, dx$
$=-cos(10)--cos(4)$
$=.185427908213$

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

$$\sum_{i=1}^{\\100} \int_4^{10}\sin(x)\,dx = f(a+i(\frac{b-a}{n}))(\frac{b-a}{n})$$\
Where b= 10, a=4, n=100, and i=1
Thefore by pluggin this into Desmos we get that the right Reimann Sums = 
$$0.191755718035$$

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

$$\sum_{i=1}^{\\100} \int_4^{10}\sin(x)\,dx = (f(a+i(\frac{b-a}{n})-(\frac{b-a}{n}))(\frac{b-a}{n})$$

Where b= 10, a=4, n=100, and i=1
Thefore by pluggin this into Desmos we get that the left Reimann Sums = 
$$0.17898883497$$

__Problem 4.__

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

Our Trapezoidal value is equal to $\frac{L_{100}+R_{100}}{2}$.
We have already found our two needed values in Problem 2 and 3 so we can just plug them into to get the trapezoidal reimann sum of

$$0.185372276502$$
__Problem 5.__
Approximate $\int_4^{10}\sin(x)\,dx$ by evaluating $M_{100}$.
$$\sum_{i=1}^{100}f\left(a+\frac{i\left(b-a\right)}{n}-\frac{\left(b-a\right)}{2n}\right)\left(\frac{\left(b-a\right)}{n}\right)$$

Where b= 10, a=4, n=100, and i=1

Thefore by pluggin this into Desmos we get that the middle Reimann Sums = 
$$0.18545572532$$

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

In using Simpsons reule we need the values of $M_{100}$ and $T_{100}$
$$S_{200}= \frac{2M_{100}+T_{100}}{3}$$
These values we have found in problems 4 and 5 and we can just substitute into the equation to get 
$$0.185427909$$















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