1. Square the Radius: The radius is the function $f(x)$. You must square it: $[f(x)]^2$.
2. The Signature of the Circle: Always place $\pi$ in front of your integral.
3. Define the Bounds: Find where the area starts ($a$) and ends ($b$) on the Axis of Rotation.
4. The Sum: Volume = $\pi \int_{a}^{b} [f(x)]^2 dx$.
Find the volume of the solid generated by rotating the region under $f(x)$ about the X-axis.
$f(x) = x^2$ from $x=0$ to $x=2$.
$f(x) = \sqrt{x}$ from $x=0$ to $x=4$.
$f(x) = 3$ (a flat line) from $x=0$ to $x=5$.
Note: This creates a Cylinder!
In the "Cylinder" problem, use the basic geometry formula $V = \pi r^2 h$. Does your integral give the same answer? Why is the integral just a fancy way of doing the same thing for curves?
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The Exponential Dome: $f(x) = e^x$ from $x=0$ to $x=1$.
The Sine Cap: $f(x) = \sqrt{\sin x}$ from $x=0$ to $x=\pi$.
The Chalice: $y = x^2$ from $y=0$ to $y=4$ rotated around the Y-axis.
1. Solve for $x$: $x = \sqrt{y}$.
2. Radius $r = \sqrt{y} \implies r^2 = y$.
3. Set up the integral with $dy$.
A semi-circle is given by $y = \sqrt{r^2 - x^2}$.
If we rotate this around the X-axis from $-r$ to $+r$, it creates a perfect **Sphere**.
Task: Use the Disk Method to prove that the volume of a sphere is $\frac{4}{3}\pi r^3$.
Objective: Explain the Disk Method to a younger sibling using a flashlight and a spinner.
The Activity:
1. Tape a flat paper shape to a pencil.
2. Spin the pencil fast.
3. Ask: "What solid shape do you see?"
The Lesson: "Our life is like that flat paper. But when we spin around God's truth, we look solid and strong. Integration adds up all the 'Circles' of our spin."
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