Section 5.6
Definition of Definite Integral
The Integral isn't just a static area; it's a dynamic accumulator. It sums up change over time to give us the total net change.
1
Net Signed Area
Area above the x-axis is Positive. Area below is Negative. The Definite Integral is the sum of these signed areas.
Net Signed Area
Positive
+1.000
Negative
−0.000
Net Area
1.000
2
Worked Example: Geometric Eval
Using Geometry (No Calculus!)
Evaluate .
1. Identify Shape
. This is a circle radius 2. The root means top half only.
2. Area Formula
3. Calculate
3
Level Up Examples
Example A: Net Signed Area
Evaluate geometrically.
1. Graph:
Line crosses x-axis at .2. Shapes:
Triangle 1 (0 to 2): Width 2, Height -4. Area = -4.Triangle 2 (2 to 4): Width 2, Height 4. Area = 4.
3. Sum:
.Example B: Splitting Integrals
Given and , find .
1. Property:
.2. Solve:
.5
Practice Quiz
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