Blog #21

This week we learned a few new things, but we also went over the Fundamental Theorem of Calculus and such.
So, I’m just going to go over Integrals and the Fundamental Theorem of Calculus.
*The integral symbol will be represented by “S” and the bounds will be represent as “3S1”.

Integrals:

First of all, an antiderivative is a general solution, while an integral is more specific where you must solve for c.
Second, an antiderivative is just the derivative backwards. You will be given f’(x) and asked to find f(x).
Third, how to find an antiderivative: If your given ax^b (a being a constant and b being an exponent), your formula is (a/b+1)(x^b+1).
*Note: When solving for the antiderivative, always add “+ c” in place of a constant. When solving for an integral, you will be given something like “f(1) = 2”, in that case, plug in 1 for x into your antiderivative and set it equal to 2 and that’s your constant.
**Note: When given the second derivative, just solve twice.
***Note: When given an integral, it is always used with the integral symbol in front (“S”) and dx afterwards.
****Note: Shortcut: If you have a fractional exponent, add one and multiply by the reciprocal.
*****Note: Trig functions: You just do the opposite of the derivative.

Next, the Fundamental Theorem of Calculus:
This theorem is very simple; it just states that “bSa f(x) dx = F(b) – F(a) = area where F(x) is the integrated function”.

Examples: Find the area of the shape between the boundaries given.

Ex. 1) 4S1 6 dx
F(x) = 6x
Area = 6(4) – 6(1) = 24 – 6
= 18

Ex. 2) 2S0 (x + 1) dx
F(x) = ½ x² + x
Area = (½ (2)² + (2)) – (½ (0)² + (0)) = 4 – 0
= 4

Ex. 3) 3S3 2x dx
= 0
**Whenever the upper and lower bounds are the same, the area is always 0.