Reimann Sum - Midpoint Approximation

[Jason76]

Top is x

Bottom is f(x)

1	2	3	4	5
15	10	9	6	5

Given

$\displaystyle \int^{5}_{1} f(x) dx$

interval: $\displaystyle [1,5]$

Using two sub-intervals of equal length

What is the midpoint Reimann approximation? Hint

 

[Jason76]

This wasn't isn't too hard to figure out

$\displaystyle \int^{5}_{1} f(x) dx$

interval: $\displaystyle [1,5]$

$\displaystyle n = 2$ - because it wants two sub-intervals

$\displaystyle a = 1$

$\displaystyle b = 5$

We use

$\displaystyle \Delta(x) =$ width $\displaystyle = \dfrac{b - a}{n} = \dfrac{5 - 1}{2} = 2$

Now we know that each interval is $\displaystyle 2$. So now we need to find the midpoint of each interval using the formula: $\displaystyle M_{1} = \dfrac{L_{1} + R_{2}}{2}$

We know what $\displaystyle L_{1}$ and $\displaystyle R_{2}$ are for each interval, by looking at a graph we will draw.

Now you would have $\displaystyle (x_{1}), (x_{2})$ which represent each midpoint.

Finally, to find the area under the curve:

$\displaystyle \Delta(x)[f(x_{1}) + f(x_{2})] =$ area

Look right

 

[srmichael]

This wasn't isn't too hard to figure out

$\displaystyle \int^{5}_{1} f(x) dx$

interval: $\displaystyle [1,5]$

$\displaystyle n = 2$ - because it wants two sub-intervals

$\displaystyle a = 1$

$\displaystyle b = 5$

We use

$\displaystyle \Delta(x) =$ width $\displaystyle = \dfrac{b - a}{n} = \dfrac{5 - 1}{2} = 2$

Now we know that each interval is $\displaystyle 2$. So now we need to find the midpoint of each interval using the formula: $\displaystyle M_{1} = \dfrac{L_{1} + R_{2}}{2}$

We know what $\displaystyle L_{1}$ and $\displaystyle R_{2}$ are for each interval, by looking at a graph we will draw.

Now you would have $\displaystyle (x_{1}), (x_{2})$ which represent each midpoint.

Finally, to find the area under the curve:

$\displaystyle \Delta(x)[f(x_{1}) + f(x_{2})] =$ area

Look right

Looks good. So what did you get for your $\displaystyle (x_{1}), (x_{2})$?

 

[Jason76]

Looks good. So what did you get for your $\displaystyle (x_{1}), (x_{2})$?

The book answer is 32. I will try to figure this one out, step by step, soon.