Double and Triple Integrals: Jacobian Transformation and Volume Computations

Double and Triple Integrals

Jacobian Transformation and Volume Computations

---

Photo by Rick Rothenberg on Unsplash

• Introduction to Multiple Integrals
• Double Integrals Basics
• Change of Variables with Jacobian
• Triple Integrals
• Volume Computations with Integrals
• Key Takeaways

---

Photo by Vitaly Gariev on Unsplash

1

Double Integrals

Fundamentals and Setup

---

Photo by Claudio Guglieri on Unsplash

• ∬R f(x,y) dA integrates function f over 2D region R
• Represents volume under surface z = f(x,y) over R
• Computed as iterated integrals: ∫a^b [∫_{g(x)}^{h(x)} f(x,y) dy] dx
• Order of integration: dy dx or dx dy depending on region

• Shaded region R in xy-plane
• Surface z = f(x,y)
• Volume approximated by Riemann sums → exact double integral
• Common for area, mass, probability

---

Photo by Shubham Dhage on Unsplash

2

Change of Variables

Using the Jacobian Determinant

---

Photo by Pavel Neznanov on Unsplash

• Transform: x = x(u,v), y = y(u,v)
• dA = |det(∂(x,y)/∂(u,v))| du dv
• Jacobian J = ∂x/∂u ∂y/∂v - ∂x/∂v ∂y/∂u
• New integral: ∬_S f(x(u,v), y(u,v)) |J| du dv
• Simplifies awkward regions (e.g., polar: r dr dθ)

3

Triple Integrals

Integrating over Volumes

• ∭E f(x,y,z) dV integrates over 3D region E
• For volume: ∭E 1 dV = volume of E
• Iterated form: e.g., ∫a^b ∫{g(x)}^{h(x)} ∫_{u(x,y)}^{v(x,y)} f dz dy dx
• Jacobian for 3 vars: |∂(x,y,z)/∂(u,v,w)|

• ∭_E 1 dV = total volume
• Divide E into small boxes of volume ΔV
• Riemann sum: Σ 1 * ΔV → ∭ 1 dV
• Spherical coords for spheres: ρ² sin φ dρ dφ dθ

---

Photo by Shubham Dhage on Unsplash

Double & triple integrals compute volumes, masses, etc. Use Jacobian |J| for coordinate changes Iterated integrals handle limits carefully

Practice with polar, cylindrical, spherical coordinates for efficiency

---

Photo by khaled khazna on Unsplash