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Ishrat Saboor & Jennifer Lupercio
There are three types of sums
Δx= b-a/n start-stop/# of rect.
In order to find the left sum you take the left side of each of the rectangles lengths and apply it to the equation and then add the sum together Δx(f(x₀)+f(x₁)+f(x₂)...)
To find the right sum you take the right side of the rectangles lengths and plug it into the equation and then add that sum up to get the right sum Δx(f(x₁)+f(x₂)+f(x₃)...)
In order to get he mid-point sum youre pretty much averaging the two points.
Reimann Sums continued...
Now here's an example...
f(x) = x2+10 ; [1,4] ; Δ=1
Left sum: f(1)+f(2)+f(3)
Right sum: f(2)+f(3)+f(4)
mid-point: 1+4/2= 2.5
Riemann Sum continued...
first you put in the equation in
In order to do the trapezoidal rule you first need to know the area formula of a trapezoid.
Here a= the first base, b= the second base, then both are divided by 2 and then multiplied by h which is the height and then getting the answer which is the area for the trapezoid.
Trapezoidal rule continued...
Now heres an example
f(x)=x^2+1 ; [0,3] ; Δ=1
left sum: 1(1)+2(1)+5(1)=1
right sum: 2(1)+5(1)+10(1)= 17
In order to do Simpsons rule you need to first know the formula.
In other words we can simply this formula by grouping to get
This make it an easier way to remember the formula
Simpson's Rule cont'd
Important!!!: Make sure n is even or else you cannot do simpson's rule
First find delta x
Then find each n
Then apply the formula
And then the answer
Next we will try this the interval of 1 to 2
Integral Calculus Examples
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Ishrat Saboor | May 23, 2016