1) Introduction: Moving from 2D to 3D 📐
We’ve mastered calculating areas with integrals, but now it’s time to step into the third dimension! This topic introduces the disk, washer, and shell methods to find the volume of solids formed by rotating regions around an axis. These techniques are like sculpting with math, turning 2D shapes into 3D objects!
We’ll explore:
- Disk Method: Volume from a solid rotation with circular cross-sections.
- Washer Method: Accounting for hollow spaces in rotations.
- Shell Method: Using cylindrical shells for a different approach.
Quick Recap: Definite integrals give areas; now we’ll use them for volumes.
2) Disk Method 🎓
The disk method calculates the volume of a solid formed by rotating a region around an axis, where each cross-section is a disk. Imagine spinning a curve around the x-axis to create a solid cylinder!
Definition 14.1: Disk Method
The volume of a solid rotated around the x-axis is \( V = \pi \int_a^b [f(x)]^2 \, dx \), where \( f(x) \) is the radius of the disk.
Example 1: Rotating a Line
Find the volume when \( y = 2x \) from 0 to 1 is rotated around the x-axis.
- Radius = \( 2x \), Volume = \( \pi \int_0^1 (2x)^2 \, dx = \pi \int_0^1 4x^2 \, dx \).
- Antiderivative: \( \frac{4x^3}{3} \).
- Evaluate: \( \pi \left[ \frac{4x^3}{3} \right]_0^1 = \pi \left( \frac{4}{3} - 0 \right) = \frac{4\pi}{3} \).
Answer: Volume = \( \frac{4\pi}{3} \) cubic units.
Example 2: Quadratic Rotation
Find the volume when \( y = x^2 \) from 0 to 2 is rotated around the x-axis.
- Radius = \( x^2 \), Volume = \( \pi \int_0^2 (x^2)^2 \, dx = \pi \int_0^2 x^4 \, dx \).
- Antiderivative: \( \frac{x^5}{5} \).
- Evaluate: \( \pi \left[ \frac{x^5}{5} \right]_0^2 = \pi \left( \frac{32}{5} - 0 \right) = \frac{32\pi}{5} \).
Answer: Volume = \( \frac{32\pi}{5} \) cubic units.
3) Washer Method 📐
The washer method is used when the solid has a hole, like a donut. It calculates the volume by subtracting the inner disk from the outer one.
Definition 14.2: Washer Method
The volume is \( V = \pi \int_a^b [R(x)^2 - r(x)^2] \, dx \), where \( R(x) \) is the outer radius and \( r(x) \) is the inner radius.
Example 3: Hollow Cylinder
Find the volume between \( y = x^2 \) and \( y = x \) from 0 to 1 around the x-axis.
- Outer radius = \( x \), Inner radius = \( x^2 \).
- Volume = \( \pi \int_0^1 (x^2 - (x^2)^2) \, dx = \pi \int_0^1 (x^2 - x^4) \, dx \).
- Antiderivative: \( \frac{x^3}{3} - \frac{x^5}{5} \).
- Evaluate: \( \pi \left[ \frac{x^3}{3} - \frac{x^5}{5} \right]_0^1 = \pi \left( \frac{1}{3} - \frac{1}{5} \right) = \pi \cdot \frac{2}{15} \).
Answer: Volume = \( \frac{2\pi}{15} \) cubic units.
Example 4: Annular Region
Find the volume between \( y = \sqrt{x} \) and \( y = x^2 \) from 0 to 1 around the x-axis.
- Outer radius = \( \sqrt{x} \), Inner radius = \( x^2 \).
- Volume = \( \pi \int_0^1 ((\sqrt{x})^2 - (x^2)^2) \, dx = \pi \int_0^1 (x - x^4) \, dx \).
- Antiderivative: \( \frac{x^2}{2} - \frac{x^5}{5} \).
- Evaluate: \( \pi \left[ \frac{x^2}{2} - \frac{x^5}{5} \right]_0^1 = \pi \left( \frac{1}{2} - \frac{1}{5} \right) = \pi \cdot \frac{3}{10} \).
Answer: Volume = \( \frac{3\pi}{10} \) cubic units.
4) Shell Method 🔍
The shell method uses cylindrical shells to find volume, especially useful when rotating around the y-axis or when the disk/washer method is complex.
Definition 14.3: Shell Method
The volume is \( V = 2\pi \int_a^b x f(x) \, dx \) when rotating around the y-axis, where \( x \) is the radius and \( f(x) \) is the height.
Example 5: Shell Around y-axis
Find the volume when \( y = x^2 \) from 0 to 1 is rotated around the y-axis.
- Radius = \( x \), Height = \( x^2 \).
- Volume = \( 2\pi \int_0^1 x \cdot x^2 \, dx = 2\pi \int_0^1 x^3 \, dx \).
- Antiderivative: \( \frac{x^4}{4} \).
- Evaluate: \( 2\pi \left[ \frac{x^4}{4} \right]_0^1 = 2\pi \left( \frac{1}{4} - 0 \right) = \frac{\pi}{2} \).
Answer: Volume = \( \frac{\pi}{2} \) cubic units.
Example 6: Complex Shape
Find the volume when \( y = \sqrt{x} \) from 0 to 4 is rotated around the y-axis.
- Radius = \( x \), Height = \( \sqrt{x} \).
- Volume = \( 2\pi \int_0^4 x \cdot \sqrt{x} \, dx = 2\pi \int_0^4 x^{3/2} \, dx \).
- Antiderivative: \( \frac{2x^{5/2}}{5} \).
- Evaluate: \( 2\pi \left[ \frac{2x^{5/2}}{5} \right]_0^4 = 2\pi \left( \frac{2 \cdot 4^{5/2}}{5} - 0 \right) = 2\pi \cdot \frac{32}{5} = \frac{64\pi}{5} \).
Answer: Volume = \( \frac{64\pi}{5} \) cubic units.
5) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Compute the volume of the solid using the disk, washer, or shell method. 📚
\( y = x \) from 0 to 2 around the x-axis (disk)
\( y = x^2 \) from 0 to 1 around the x-axis (disk)
\( y = \sqrt{x} \) from 0 to 4 around the x-axis (disk)
\( y = x^2 \) and \( y = x \) from 0 to 1 around the x-axis (washer)
\( y = x^3 \) from 0 to 1 around the y-axis (shell)
\( y = 2 - x \) from 0 to 2 around the x-axis (disk)
\( y = x^2 \) and \( y = 2x \) from 0 to 1 around the x-axis (washer)
\( y = \sqrt{x} \) from 0 to 9 around the y-axis (shell)
\( y = x^2 + 1 \) from 0 to 2 around the x-axis (disk)
\( y = x \) and \( y = x^2 \) from 0 to 1 around the y-axis (shell)
\( y = \sin(x) \) from 0 to \( \pi \) around the x-axis (disk)
Challenging Practice Questions 🌟
Instructions: These involve complex rotations or multiple methods. 🧠
Find the volume of \( y = x^3 \) from 0 to 2 rotated around the x-axis using both disk and shell methods.
Compute the volume between \( y = x^2 \) and \( y = 2x - x^2 \) from 0 to 1 around the x-axis (washer).
Determine the volume of \( y = e^x \) from 0 to 1 around the y-axis (shell).
Evaluate the volume between \( y = \cos(x) \) and \( y = 0 \) from 0 to \( \pi/2 \) around the x-axis (disk).
Find the volume of \( y = 1/x \) from 1 to 2 rotated around the x-axis, considering the washer method.
6) Summary & Cheat Sheet 📋
6.1) Disk Method
\( V = \pi \int_a^b [f(x)]^2 \, dx \) for rotation around x-axis.
6.2) Washer Method
\( V = \pi \int_a^b [R(x)^2 - r(x)^2] \, dx \) for hollow solids.
6.3) Shell Method
\( V = 2\pi \int_a^b x f(x) \, dx \) for rotation around y-axis.
You’ve conquered volumes! Next, we’ll tackle improper integrals. 🎉