Attempting to Teach Myself Calculus - Sections 5.1-5.2

I wanted to write an entry, or entries, on the things I'm teaching myself for Calculus I as a way to keep track, and as perhaps a mild source of entertainment for others. I'm starting with section 5.1, and then talking about section 5.2.

Section 5.1


The section started with the definition of an antiderivative. The definition is as follows:

The Definition of Antiderivative
A function F is an antiderivative of f on an interval I if F'(x)=f(x) for all x in I.

So, basically, a function is an antiderivative if it's derivative equals another function.

Next, it talked about Theorem 5.1, which is as follows:

Representation of Antiderivatives
If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x)=F(x)+C, for all x in I, where C is a constant.

When you take the derivative of a function, any stand-alone constant is written as 0. So, since you don't know whether there is a constant or not, you have to assume that there is some constant, C, that needs to be added.

Notation for Antiderivatives


The section also talked about how differentiation is the inverse of integration. So, if you have found the integral, you can use differentiation to return to the original function.

Basic Integration Rules


So what I gathered from this section is that finding antiderivatives is just like going backward from what I had been doing previously when finding derivatives, and you must add some constant, C, while figuring it ouit since you don't know whether there is some stand-alone constant or not.

Section 5.2


The section started out with an explanation of sigma (∑) notation. It says the following:

The sum of n terms a₁, a₂, a₃, . . . a_n is written as


where i is the index of summation, a_i is the ith term of the sum, and the upper and lower bounds of summation are n and 1.

Then, the book gave me summation formulas. They are as follows:


Next, they discussed the area of a plane region. In order to find the area, you separate the area you want to find into a series of rectangles. Once, with the rectangles larger than the area of what you're trying to find, and once with the rectangles smaller than the area you are trying to find. Then, you find the sum of the area of the rectangles by multiplying f(the height) times the width, and using summation to find the result. That way, you have an upper and lower bounds by which to come up with the actual area of the region.


The section said that if you find the limit as n approaches infinity (basically making the number of rectangles that separate the area up infinite) of the sum of f(m_i) (the minimum bounds) times ∆x [which is (b-a)/n] and the same of the maximum bounds, you will find that they are equal and are the area of the region.

It is summed up in this:
Defintion of the Area of a Region in the Plane
Let f be continuous and nonnegative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is

where ∆x = (b-a)/n.


That's basically it from those sections. I'll write again on sections 5.3-5.7 when I get there.