#### *Related to [[Antiderivatives]] and [[Integration]]*
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## Numerical approximation
- There are some situations where approximating an integral is more advantageous then evaluating it exactly. To *approximate* an integral we use **Reimann sums, which are outlined in detail on the [[Integration]] page**. This section will discuss some more advanced **Reimann sum techniques that allow us to approximate integrals more accurately.**
##### *Method 1: the midpoint rule*
- The midpoint rule is **the second most accurate Reimann sum technique**. It is **always more accurate than the trapezoid rule** and far better than the left/right hand rule.
- *Recall the left/right hand rule sets the height of each rectangle in the Reitman sum equal to the height of the function at its upper left or right hand corner.*
- The midpoint rule relies on using multiple rectangles that have a height equal to the value of the **evaluated function at their midpoint *(referring to the rectangles)*.**
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$\text{For }F(x),\text{ integer }n,\;x_i=a+\frac{b-a}{n}\cdot i$
$\int_a^b f(x)\,dx =\sum_{i=1}^n\,f\biggl(\frac{x_{i-1}+x_i}{2}\biggl)\,\cdot\,\frac{b-a}{n}\;\;\;{\color{grey}\longrightarrow}\;\;\Delta x\Biggl[f\Big(x_1-\frac{\Delta x}{2}\Big)+f\Big(x_2-\frac{\Delta x}{2}\Big)...\;\Biggl]$
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##### *Method 2: the trapezoid rule*
- The trapezoid rule uses multiple trapezoids instead of rectangles to approximate the area under a curve.
- When using the trapezoid rule you multiply every output by 2 except for the **first and last outputs evaluated**.
- You will always end up evaluating one more output than the number given by **n** because you have **to evaluate the beginning AND end of every section not just the left or right side.**
- The trapezoid rule is less accurate than the midpoint rule and Simpson's rule, but more accurate than the left/right endpoint rules.
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$\text{For }F(x),\text{ integer }n,\;x_i=a+\frac{b-a}{n}\cdot i$
$\int_a^bf(x)\,dx=\frac{b-a}{n}\cdot\frac{1}{2}\biggl(f(x_0) +2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)...+f(x_{last})\biggl)$
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##### *Method 3: Simpson's rule*
- Simpson's rule uses parabolas to approximate integrals. As you can see in the generalized formula below, **the first and last height sampled should be multiplied by one, every other height *(y value)* should alternate between being multiplied by 4 and 2.** As a result of using parabolas Simpson's rule does not use right, left or middle points for its $x$ values, instead it evaluates the function **at the right AND left endpoints of every section** just like the trapezoid rule.
- Approximations using Simpson's rule are equal to the **weighted** average of the approximations you would get from the midpoint and trapezoidal rules. Simpson's rule **is by far the most accurate of any approximation method.** $S_n=\frac{1}{3}T_n+\frac{2}{3}M_n$
- ==n MUST be even when using Simpson's rule.==
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$\int_a^bf(x)\,dx=\frac{b-a}{n}\cdot \frac{1}{3}\Big[f(x_1)+4f(x_2)+2f(x_3)+4f(x_4)+2f(x_5)... + f(x_{last})\Big]$
$\text{let }x_1=a\text{ and }x_{last}=b\;\;\;\;\;\;\text{let the number of heights evaluated }=n+1$
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- **==Its important to remember that when using Simpson's rule you MUST evaluate the function at both the beginning AND end of every *"section"* specified by $n$. This means if n = 2 you will actually evaluate 3 height values, at the begging, middle, and end of the bound because 3 lines are needed to make 2 rectangular sections.==** For example, if the bounds are 0-1 and n = 2 your sum will look like this:
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$\frac{0.5}{3}\Big[f(0)+4f(0.5)+f(1)\Big]$
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##### *Calculating error*
- The variable $E_M$ is the **value** error between the actual value of the integral and the value you get using $n$ rectangles and the **midpoint Reimann sum technique.** Likewise $E_T$ represents the error between the actual value of the integral and the value you get using $n$ rectangles and the **trapezoid Reimann sum technique**.
- Note the values you get from these equations represent the **maximum possible error, the actual value error could be less but NEVER more**.
- If the derivative of an equation is equal to zero, when solving for k **that shows that your approximation is *perfectly accurate*.
- The value of $k$ in the error equation for ==the midpoint and trapezoidal techniques== is any value that is larger than **all outputs on the specified bounds of the function $f(x)