Average Value of a Function
How do you find the average height of a curve over an interval? You "smush" the area into a rectangle.
"Integral divided by Interval"
Find the average value of f(x) = x^2 on [0, 3].
- Interval Width: b - a = 3 - 0 = 3.
- Integrate: ∫₀³ x² dx = [x³/3]₀³ = 27/3 - 0 = 9.
- Divide: Average = 9 / 3 = 3.
Position, Velocity, Acceleration (Integrals)
In Unit 4, we differentiated position to get velocity. Now, we integrate velocity to get position.
| Concept | Formula | Meaning |
|---|---|---|
| Displacement | ∫ v(t) dt | Net change in position. (Can be 0 if you return home). |
| Total Distance | ∫ |v(t)| dt | Total ground covered. (Odometer reading). |
| Current Position | s(0) + ∫₀ᵗ v(x) dx | Start Position + Displacement. |
Area Between Curves
To find the area between two functions, we slice it into rectangles. Height = Top - Bottom.
[Image of area between two curves]Vertical Slices (dx)
Top Function - Bottom Function
Horizontal Slices (dy)
Right Function - Left Function
Volumes with Cross Sections
Imagine a 3D object built on a base area R. We find the volume by integrating the Area of the Slice.
If the base of the slice is s = Top - Bottom:
- Square: A = s^2
- Semicircle: A = (π/8)s^2
- Equilateral Triangle: A = (√3/4)s^2
- Rectangle: A = s · h (Height usually given).
Rotational Volume: The Disc Method
Used when the region is flush against the axis of rotation (no gap).
Think: Sum of areas of circles (πr^2).
R is the distance from the curve to the axis of rotation.
Rotational Volume: The Washer Method
Used when there is a gap between the region and the axis of rotation. The slice looks like a washer (donut).
"Big Radius Squared minus Little Radius Squared"
Distance from Outer Curve to Axis.
Distance from Inner Curve to Axis.
Warning: Calculate R^2 - r^2. Do NOT calculate (R - r)^2. That is Algebra murder.