Download Integrals - San Diego Unified School District

Bernard Nefalar
Calculus
Period 4
Introduction
5/31/2013
Outcome: Students will be able to evaluate integrals.
Warm Up
Standards
1
10+5=15 mins
Find dy.
dx
1) y + 7x6 = 0
2) xy - 3x5 = -1
3) y2 - 13x4 + 6x3 + x = – 1
x
Topic 1: Integrals
Definition
Antiderivative (Integral) – F(x): A function x of y such that
x’ = y. (Think opposite of derivative.)
[Don’t write this.]
Question…What is the difference between
dy of 2x3 + 5
and
dy of 2x3 + 99?
dx
dx
Does the 5 or 99 matter? But the functions are totally
different.
Here we introduce C or the constant. This is important for
antiderivatives/integrals. In addition, each f’(x) will have a
dx attached to it which comes from the dx of the dy.
dx
Given f’(x) then,
f(x) + C =
f’(x) dx.
(9:25-9:40) 30 mins.
Procedure
Step 1: Start with the exponent and add 1 to find
the new exponent.
Step 2: Divide the coefficient with this new
exponent.
Step 3: Don’t forget to add the C (constant to the
new function to cover all of the constants
that will make the function true).
Note:
Given a f(x) multiplied by a constant c,
cf(x) dx = c f(x) dx.
For examples 1-6, find the antiderivative.
Example 1
x2 dx
Example 2
(x2 – 5) dx
Example 3
5 dx
Example 4
1 dx
x3
Example 5
5(x2 + 1) dx
Example 6
-(x3 + 5x) dx
Checkpoint
Find the antiderivative.
1) (x2 – 2) dx
2) 3(x2 – 2x + 3) dx
3) (x3/2 + 2x + 1) dx 4) 1 dx
2x4
(9:40-9:55) 15+5=20 mins.