**Integral Calculus**

Ishrat Saboor & Jennifer Lupercio

Ishrat Saboor & Jennifer Lupercio

Riemann Sums

Trapezoidal Rule

Simpson's Rule

There are three types of sums

- left sum
- right sum
- mid-point sum

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)

(11)+(14)+(!9)= 44ft

Right sum: f(2)+f(3)+f(4)

(14)+(19)+(26)=59ft

mid-point: 1+4/2= 2.5

Riemann Sum continued...

example #2

f(x)= x^3+8

first you put in the equation in

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.

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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

trap. rule:

1+2/2(1)+2+5/2(1)+5+10/2(1)

(1.5)+(3.5)+(7.5)= 12.5

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

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Important!!!: Make sure n is even or else you cannot do simpson's rule

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First find delta x

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Then find each n

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Then apply the formula

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And then the answer

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Next we will try this the interval of 1 to 2

Integral Calculus Examples

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