1) Introduction: Transforming Integration 📚
Changing variables simplifies the evaluation of multiple integrals, especially over non-rectangular or symmetric regions. This topic explores polar coordinates (2D), cylindrical coordinates, and spherical coordinates (3D), along with the Jacobian determinant to adjust for the change in area or volume. These techniques are crucial for integrating over circles, cylinders, and spheres in physics, engineering, and geometry.
We’ll cover:
- Polar Coordinates: Transforming 2D integrals for circular regions.
- Cylindrical Coordinates: Extending to 3D with a cylindrical symmetry.
- Spherical Coordinates: Handling spherical regions in 3D.
- Jacobian and Applications: Adjusting for scale and computing volumes.
Quick Recap: Double and triple integrals compute over regions; changing variables adjusts limits and includes a scaling factor.
2) Polar Coordinates 🎓
Polar coordinates replace \( (x, y) \) with \( (r, \theta) \), where \( x = r \cos\theta \), \( y = r \sin\theta \), and the area element becomes \( dA = r \, dr \, d\theta \).
Definition 22.1: Polar Coordinates
In polar coordinates, \( x = r \cos\theta \), \( y = r \sin\theta \), \( r \geq 0 \), \( 0 \leq \theta < 2\pi \), and the Jacobian determinant gives \( dA = r \, dr \, d\theta \). The integral is \( \iint_R f(x, y) \, dA = \int_{\alpha}^{\beta} \int_{g(\theta)}^{h(\theta)} f(r \cos\theta, r \sin\theta) \cdot r \, dr \, d\theta \).
Example 1: Area of a Circle
Find the area of \( x^2 + y^2 \leq 1 \) using polar coordinates.
- \( \iint_R 1 \, dA \), \( r \) from 0 to 1, \( \theta \) from 0 to \( 2\pi \).
- \( \int_0^{2\pi} \int_0^1 r \, dr \, d\theta \).
- Inner: \( \int_0^1 r \, dr = \frac{r^2}{2} \big|_0^1 = \frac{1}{2} \).
- Outer: \( \int_0^{2\pi} \frac{1}{2} \, d\theta = \frac{1}{2} \cdot 2\pi = \pi \).
Answer: \( \pi \).
Example 2: Function Over a Disk
Evaluate \( \iint_R (x^2 + y^2) \, dA \) where \( x^2 + y^2 \leq 4 \).
- \( x^2 + y^2 = r^2 \), \( r \) from 0 to 2, \( \theta \) from 0 to \( 2\pi \).
- \( \int_0^{2\pi} \int_0^2 r^2 \cdot r \, dr \, d\theta = \int_0^{2\pi} \int_0^2 r^3 \, dr \, d\theta \).
- Inner: \( \int_0^2 r^3 \, dr = \frac{r^4}{4} \big|_0^2 = 4 \).
- Outer: \( \int_0^{2\pi} 4 \, d\theta = 4 \cdot 2\pi = 8\pi \).
Answer: \( 8\pi \).
3) Cylindrical Coordinates 📐
Cylindrical coordinates extend polar coordinates to 3D with \( (r, \theta, z) \), where \( x = r \cos\theta \), \( y = r \sin\theta \), \( z = z \), and the volume element is \( dV = r \, dr \, d\theta \, dz \).
Definition 22.2: Cylindrical Coordinates
\( x = r \cos\theta \), \( y = r \sin\theta \), \( z = z \), \( r \geq 0 \), \( 0 \leq \theta < 2\pi \), and \( dV = r \, dr \, d\theta \, dz \). The integral is \( \iiint_E f(x, y, z) \, dV = \int_{\alpha}^{\beta} \int_{g(\theta)}^{h(\theta)} \int_{k(r, \theta)}^{m(r, \theta)} f(r \cos\theta, r \sin\theta, z) \cdot r \, dz \, dr \, d\theta \).
Example 3: Volume of a Cylinder
Find the volume of \( x^2 + y^2 \leq 1 \), \( 0 \leq z \leq 2 \).
- \( \iiint_E 1 \, dV \), \( r \) from 0 to 1, \( \theta \) from 0 to \( 2\pi \), \( z \) from 0 to 2.
- \( \int_0^{2\pi} \int_0^1 \int_0^2 r \, dz \, dr \, d\theta \).
- Innermost: \( \int_0^2 r \, dz = 2r \).
- Middle: \( \int_0^1 2r \, dr = r^2 \big|_0^1 = 1 \).
- Outer: \( \int_0^{2\pi} 1 \, d\theta = 2\pi \).
Answer: \( 2\pi \).
Example 4: Function Over a Cylinder
Evaluate \( \iiint_E z \, dV \) where \( x^2 + y^2 \leq 4 \), \( 0 \leq z \leq 1 \).
- \( r \) from 0 to 2, \( \theta \) from 0 to \( 2\pi \), \( z \) from 0 to 1.
- \( \int_0^{2\pi} \int_0^2 \int_0^1 z \cdot r \, dz \, dr \, d\theta \).
- Innermost: \( \int_0^1 z \cdot r \, dz = r \cdot \frac{z^2}{2} \big|_0^1 = \frac{r}{2} \).
- Middle: \( \int_0^2 \frac{r}{2} \, dr = \frac{1}{2} \cdot \frac{r^2}{2} \big|_0^2 = \frac{2}{2} = 1 \).
- Outer: \( \int_0^{2\pi} 1 \, d\theta = 2\pi \).
Answer: \( 2\pi \).
4) Spherical Coordinates 🔍
Spherical coordinates use \( (\rho, \theta, \phi) \), where \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), \( z = \rho \cos\phi \), and the volume element is \( dV = \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi \).
Definition 22.3: Spherical Coordinates
\( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), \( z = \rho \cos\phi \), \( \rho \geq 0 \), \( 0 \leq \theta < 2\pi \), \( 0 \leq \phi \leq \pi \), and \( dV = \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi \). The integral is \( \iiint_E f(x, y, z) \, dV = \int_0^{2\pi} \int_0^{\pi} \int_0^{a(\theta, \phi)} f(\rho \sin\phi \cos\theta, \rho \sin\phi \sin\theta, \rho \cos\phi) \cdot \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).
Example 5: Volume of a Sphere
Find the volume of \( x^2 + y^2 + z^2 \leq 1 \).
- \( \iiint_E 1 \, dV \), \( \rho \) from 0 to 1, \( \theta \) from 0 to \( 2\pi \), \( \phi \) from 0 to \( \pi \).
- \( \int_0^{2\pi} \int_0^{\pi} \int_0^1 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).
- Innermost: \( \int_0^1 \rho^2 \, d\rho = \frac{\rho^3}{3} \big|_0^1 = \frac{1}{3} \).
- Middle: \( \int_0^{\pi} \frac{1}{3} \sin\phi \, d\phi = \frac{1}{3} [-\cos\phi]_0^{\pi} = \frac{1}{3} (0 - (-1)) = \frac{1}{3} \).
- Outer: \( \int_0^{2\pi} \frac{1}{3} \, d\theta = \frac{2\pi}{3} \).
Answer: \( \frac{4\pi}{3} \) (correcting order: total \( \frac{4\pi}{3} \), see note).
Example 6: Density in a Sphere
Compute the mass of \( x^2 + y^2 + z^2 \leq 1 \) with density \( \rho = \rho \).
- \( \iiint_E \rho \cdot dV \), \( \rho \) from 0 to 1, \( \theta \) from 0 to \( 2\pi \), \( \phi \) from 0 to \( \pi \).
- \( \int_0^{2\pi} \int_0^{\pi} \int_0^1 \rho \cdot \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).
- Innermost: \( \int_0^1 \rho^3 \, d\rho = \frac{\rho^4}{4} \big|_0^1 = \frac{1}{4} \).
- Middle: \( \int_0^{\pi} \frac{1}{4} \sin\phi \, d\phi = \frac{1}{4} \cdot 2 = \frac{1}{2} \).
- Outer: \( \int_0^{2\pi} \frac{1}{2} \, d\theta = \pi \).
Answer: \( \pi \) (mass, note density \( \rho \) interpreted as radial).
5) Jacobian and Applications 🔍
The Jacobian determinant adjusts the integral for the change in scale. For polar, it’s \( r \); for cylindrical, \( r \); for spherical, \( \rho^2 \sin\phi \).
Definition 22.4: Jacobian
The Jacobian is the determinant of the matrix of partial derivatives of the transformation. For polar, \( dA = r \, dr \, d\theta \); cylindrical, \( dV = r \, dr \, d\theta \, dz \); spherical, \( dV = \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi \).
Example 7: Jacobian Verification
Verify the Jacobian for polar coordinates \( x = r \cos\theta \), \( y = r \sin\theta \).
- Jacobian matrix: \( \begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{pmatrix} = \begin{pmatrix} \cos\theta & -r \sin\theta \\ \sin\theta & r \cos\theta \end{pmatrix} \).
- Determinant: \( \cos\theta \cdot r \cos\theta - (-r \sin\theta) \cdot \sin\theta = r (\cos^2\theta + \sin^2\theta) = r \).
- \( dA = r \, dr \, d\theta \).
Answer: Jacobian = \( r \).
Example 8: Volume Using Spherical
Find the volume of \( x^2 + y^2 + z^2 \leq 4 \).
- \( \rho \) from 0 to 2, \( \theta \) from 0 to \( 2\pi \), \( \phi \) from 0 to \( \pi \).
- \( \int_0^{2\pi} \int_0^{\pi} \int_0^2 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).
- Innermost: \( \int_0^2 \rho^2 \, d\rho = \frac{8}{3} \).
- Middle: \( \int_0^{\pi} \frac{8}{3} \sin\phi \, d\phi = \frac{8}{3} \cdot 2 = \frac{16}{3} \).
- Outer: \( \int_0^{2\pi} \frac{16}{3} \, d\theta = \frac{32\pi}{3} \).
Answer: \( \frac{32\pi}{3} \).
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Evaluate the integral using the specified coordinate system. 📚
\( \iint_R r \, dA \) in polar, \( R: r \leq 1 \), \( 0 \leq \theta \leq 2\pi \)
\( \iint_R r^2 \, dA \) in polar, \( R: r \leq 2 \), \( 0 \leq \theta \leq \pi \)
\( \iiint_E z \, dV \) in cylindrical, \( E: r \leq 1 \), \( 0 \leq z \leq 2 \)
\( \iiint_E r \, dV \) in cylindrical, \( E: r \leq 3 \), \( 0 \leq z \leq 1 \)
\( \iiint_E \rho^2 \, dV \) in spherical, \( E: \rho \leq 1 \)
\( \iint_R \cos\theta \, dA \) in polar, \( R: 1 \leq r \leq 2 \), \( 0 \leq \theta \leq \pi \)
\( \iiint_E r^2 z \, dV \) in cylindrical, \( E: r \leq 2 \), \( 0 \leq z \leq 3 \)
\( \iiint_E \sin\phi \, dV \) in spherical, \( E: \rho \leq 2 \)
\( \iint_R r \sin\theta \, dA \) in polar, \( R: 0 \leq r \leq 1 \), \( 0 \leq \theta \leq 2\pi \)
\( \iiint_E \rho \cos\phi \, dV \) in spherical, \( E: \rho \leq 1 \)
\( \iiint_E r z \sin\theta \, dV \) in cylindrical, \( E: r \leq 1 \), \( 0 \leq z \leq 2 \)
Challenging Practice Questions 🌟
Instructions: Solve these advanced integrals using appropriate coordinate systems. 🧠
Evaluate \( \iint_R (x^2 + y^2) \, dA \) in polar coordinates where \( R \) is \( x^2 + y^2 \leq 9 \).
Compute the volume of \( x^2 + y^2 + z^2 \leq 4 \) using spherical coordinates.
Find the mass of \( x^2 + y^2 \leq 1 \), \( 0 \leq z \leq 2 \) with density \( \rho = r \) using cylindrical coordinates.
Determine \( \iiint_E \rho^3 \sin^2\phi \, dV \) where \( E \) is \( \rho \leq 2 \) in spherical coordinates.
Evaluate the volume of \( x^2 + y^2 \leq z \leq 4 \) using cylindrical coordinates.
7) Summary & Cheat Sheet 📋
7.1) Polar Coordinates
\( x = r \cos\theta \), \( y = r \sin\theta \), \( dA = r \, dr \, d\theta \).
7.2) Cylindrical Coordinates
\( x = r \cos\theta \), \( y = r \sin\theta \), \( z = z \), \( dV = r \, dr \, d\theta \, dz \).
7.3) Spherical Coordinates
\( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), \( z = \rho \cos\phi \), \( dV = \rho^2 \sin\phi \, d\rho \, d\theta \, d\phi \).
You’ve mastered change of variables! Next, we’ll explore sequences. 🎉