Riemann Sums

class: center, middle, inverse, title-slide

.title[
# Riemann Sums
]
.subtitle[
## Session 01
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-29
]

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# Goals:

* Introduce the problem of compute the area under a curve. 
 * Interval partitions. 
 * Riemann sums.
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# Equilibriums

> Assume that you have a string of length `$L$` and you want to hang it from a point `$P$` on the string. Where should you hang it so that the string is in equilibrium? 
If the string has a uniform density `$\rho$` and the gravitational acceleration is `$g$`, then the equilibrium point is the center of mass of the string which is at the half-point of the string.

What if the density is not uniform?

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## Let's picture this

> There is no formulae to compute the areas under general curves (except for some special cases). But we can estimate the value using some approximations via areas of rectangles.

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## Interval partitions

> __Definition:__ Let `$[a,b]$` be an interval. A partition of `$[a,b]$` is a finite set of points `$P=\{x_0,x_1,\ldots,x_n\}$` such that `$a=x_0<x_1<\ldots<x_n=b.$` 
 The length of the subinterval `$[x_{i-1},x_i]$` is denoted by `$\Delta x_i=x_i-x_{i-1}.$`

__Remark:__ Note that the length of the subintervals are not necessarily equal. In the case were all subintervals have the same length, we say that the partition is __regular__.

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#### Example

1. Let `$[a,b]=[0,1]$` and `$P=\{0,0.2,0.5,0.8,1\}.$` Then `$\Delta x_1=0.2,$` `$\Delta x_2=0.3,$` `$\Delta x_3=0.3,$` `$\Delta x_4=0.2.$`

1. Let `$[a,b]=[0,1]$` and `$P=\{0,0.25,0.5,0.75,1\}.$` Then `$\Delta x_1=0.25,$` `$\Delta x_2=0.25,$` `$\Delta x_3=0.25,$` `$\Delta x_4=0.25.$`

1. Let `$[-10,5]$` and `$P=\{-10,-5,0,5\}.$` Then `$\Delta x_1=5,$` `$\Delta x_2=5,$` `$\Delta x_3=5.$`

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#### Practice

1. Find a regular partition of `$[0,1]$` with 10 subintervals. 
1. Find a partition of `$[0,1]$` with three subintervals. 
1. Find a regular partition of `$[-1,1]$` with 5 subintervals. 
1. Find a regular partition of `$[-1,1]$` with 10 subintervals. 
1. Find a partition of `$[-10,0]$` with 6 subintervals. 
1. Find a partition of `$[-1,10]$` with 7 subintervals.

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## Riemann sums

> __Definition:__ Let `$f:[a,b]\to\mathbb{R}$` be a function and `$P=\{x_0,x_1,\ldots,x_n\}$` be a partition of `$[a,b].$` The Riemann sum of `$f$` with respect to `$P$` is defined as `$$S(f,P)=\sum_{i=1}^n f(c_i)\Delta x_i,$$` where `$c_i\in[x_{i-1},x_i].$`

__Remark:__ Note that the Riemann sum is a weighted sum of the values of the function `$f$` at the points `$c_i.$` If `$c_i=x_{i-1}$` then we call it a __left sum__, if `$c_i=x_i$` then we call it a __right sum__, and if `$c_i$` is the middle point in `$[x_{i-1},x_i]$` then we call it a __middle sum__.

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#### Example

1. Let `$f(x)=x^2$` and `$P=\{0,0.2,0.5,0.8,1\}.$` Compute the left sum, the right sum, and the middle sum. 
--
 
 * Then `$S(f,P)=0^2\cdot 0.2+0.2^2\cdot 0.3+0.5^2\cdot 0.3+0.8^2\cdot 0.2=0.3.$` 
 * Then `$S(f,P)=0.2^2\cdot 0.2+0.5^2\cdot 0.3+0.8^2\cdot 0.3+1^2\cdot 0.2=0.5.$` 
 * Then `$S(f,P)=0.1^2\cdot 0.2+0.4^2\cdot 0.3+0.7^2\cdot 0.3+0.9^2\cdot 0.2=0.41.$`

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#### Example

1. Let `$f(x)=9-x^2$` and `$P=\{0,0.25,0.5,0.75,1\}.$` Compute the left sum, the right sum, and the middle sum. 
--
 
 * Then `$S(f,P)=(9-0^2)\cdot 0.25+ (9-0.25^2)\cdot 0.25+(9-0.5^2)\cdot 0.25+(9-0.75^2)\cdot 0.25=8.78125.$` 
 * Then `$S(f,P)=(9-0.25^2)\cdot 0.25+(9-0.5^2)\cdot 0.25+(9-0.75^2)\cdot 0.25+(9-1^2)\cdot 0.25=8.53125.$` 
 * Then `$S(f,P)=(9-0.125^2)\cdot 0.25+(9-0.375^2)\cdot 0.25+(9-0.625^2)\cdot 0.25+(9-0.875^2)\cdot 0.25=8.65625.$`

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#### Practice

1. Let `$f(x)=x^2-3x+5$` over the interval `$[0,3].$` Compute the left, right and middle Riemann sum with a regular partition made of 5 subintervals.