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# Define riemann sum. Description

##### Description: These notes cover Calculus 2 classes up until this day, and it’s a summary of what’s gonna be on the midterm.
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MATH 205:We also discuss several other topics like Name a piece of the second industrial revolution.

Riemann Sum:

The area under a curve can be found in a graph by a approximation: dividing the area in rectangles and adding them up. The smaller these rectangles’s bases are, the better the approximation isDon't forget about the age old question of In order to be delegated primacy by us epa, a state must do what?

We also discuss several other topics like Who leads the independence of mexico?

If you want to learn more check out What are the three major results needed to derive trigonometrical functions?

Integrals:

Just as subtraction is the opposite way of addition, integral is of derivatives. So that means that to INTEGRATE a function, that function must be differentiable in the given boundaries. By definition:Don't forget about the age old question of What is the meaning of mutualistic interactions?

where : [a, b]We also discuss several other topics like What is the optimal temperature for each enzyme?

which  means that in order to find the area under a function, you can integrate it.

• Rules of integration:

10) ∫f(x)g1(x)dx = f(x)g(x) -  ∫f1(x)g(x)dx

* FTC :  ∫f1(x)dx = f(x)

⟶ substitution rule:  ∫f(x)g(x)dx ➝ you can set either f(x) or g(x) to be u (taking into consideration their derivatives and what’s going to be more helpful), solve for u (charging the boundaries for u (x) and plugging the function back at every u.

• tips:

➝ integration by parts: formula 10. You should use it when you know how to integrate part 2 but not part 1, that is, f’(x)g(x) becomes one function not the multiplication of two functions.

• Part 4 : left side
• Part 2 : right-side

*reminder: if you are given a definite integral, your answer should be a number, NO variables. If you are given an indefinite integral, then your answer should be w. r. t the same variable as given.

• Special case :

➝ area between curves:

where:

➝ always start by finding the intersection points (I0, I1, I2, I3)

➝ sometimes it’s necessary to invert x and y-axis, i.e. when you’re given y-values as boundaries, or when the function makes it harder to use x-values.

• Trigonometric important functions to know by heart:

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