Blog #20

This week, besides the AP review, we learned about Definite Integrals, the Fundamental Theorem of Calculus, and the Properties of Integrals.
But, I’m just going to go over the Fundamental Theorem of Calculus and the Integral Properties.
*The integral symbol will be represented by “S” and the bounds will be represent as “1S3”.

Firstly, 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.

Secondly, Integral Properties:

1. bSa f(x) dx = - (aSb) f(x) dx
2. Integrals can be broken up into many smaller integrals.
Ex. 7S0 f(x) dx = 3S0 f(x) dx + 5S3 f(x) dx + 7S5 f(x) dx
3. S k f(x) dx = k S f(x) dx where k stands for a constant.

Examples: Rewrite.

Ex. 4) 2S4 2x dx <-- Property 1
= - 4S2 2x dx

Ex. 5) 2S-1 x dx <-- Property 2
= 0S-1 x dx + 1S0 x dx + 2S1 x dx

Ex. 6) 7 6S3 2x dx <-- Property 3
= 6S3 14x dx