# Definite Integrals

**DEFINITE** means it's `defined`, means both two **boundaries** are **constant numbers**.

## `Definite integrals properties`

[Refer to Khan academy article: Definite integrals properties review](https://www.khanacademy.org/math/ap-calculus-bc/bc-accumulation-riemann-sums/modal/a/definite-integrals-properties-review)

![image](https://user-images.githubusercontent.com/14041622/40904743-47305e6a-680e-11e8-8518-1102d6922508.png)

## `Definite integral ←→ Limit of Riemann Sum`

### Example

![image](https://user-images.githubusercontent.com/14041622/41026547-6ab438ee-69a7-11e8-8f03-0e9c498d8bd5.png) Solve:

* It's easy to find `Δx = (π-0)/n = π/n`
* And `x𝖎 = S(𝖎) = a + 𝖎·Δx = 0+𝖎·Δx = 𝖎·π/n`
*   So the result is:

    ![image](https://user-images.githubusercontent.com/14041622/41026881-3a592a1e-69a8-11e8-82ad-f4be0f877d20.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41027374-6bbab04a-69a9-11e8-9402-53c1b6dc6284.png) Solve:

* Look at the boundaries, it's from `0 -> 5`,
* So the `Δx` must be cut to `n` pieces, whereas `Δx = (5-0)/n = 5/n`
* From the definition, We know the function `f(x) = x+1`
* To fill in the `x𝖎` in `f(x𝖎)`, we need to figure out the sequence:
* Sequence `x𝖎 = S(𝖎) = a+𝖎·Δx`, and since `a` represents the bottom boundary,
* So `x𝖎 = 𝖎(0+Δx) = 𝖎·5/n`
*   Get `x𝖎` back in `f(x)` to have:

    ![image](https://user-images.githubusercontent.com/14041622/41027638-217c2d00-69aa-11e8-97dd-1bbcf6d7d205.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41027746-72e50522-69aa-11e8-88b8-5bdeddcf3aec.png) Solve:

* We could easily get that the `Δx = 5/n`
* And the function is `f(x) = ln(x)`
* Since the `Δx` comes from Top & Bottom boundaries,&#x20;
* So `Δx = (Top - Bottom)/n = 5/n = (Top - 2)/n`,
*   And we get the `Top = 7`, and the Definite Integral is:

    ![image](https://user-images.githubusercontent.com/14041622/41028259-9960efc6-69ab-11e8-9f5b-4099a8976544.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41029271-dba71250-69ad-11e8-9192-28d44462d192.png) Solve:

*   See the `i=1` means it's using `Right Riemann Sum`, so the integral would be:

    ![image](https://user-images.githubusercontent.com/14041622/41029604-a5c3cb46-69ae-11e8-8bdc-c90f4a70f3f4.png)
* The `Δx = 9/n` is easily seen.
* And we need to get the Sequence `x𝖎 = S(𝖎) = a + (𝖎-1)·Δx = (𝖎-1)·9/n`
* What we got there above, tells us `a=0`.
* According to that`Δx = (b-a)/n = (b-0)/n = 9/n`, we get `b = 9`
*   So the answer is:

    ![image](https://user-images.githubusercontent.com/14041622/41029902-8f99f254-69af-11e8-9b93-22f00d9f9b67.png)
