Multiple Integration

In 1D calculus, we integrate over a line to find an area. In Multivariable Calculus, we integrate over a 2D region (Double Integral) to find a volume, or over a 3D region (Triple Integral) to find a total mass or charge.

Double Integrals: Volumes under Surfaces

The integral $\iint_R f(x, y) dA$ represents the volume between the $xy$-plane and the surface $z = f(x, y)$ over the region $R$.

Fubini’s Theorem: If the function is nice, you can calculate the double integral by doing two “nested” single integrals in any order: $\int_a^b \int_c^d f(x, y) dy \, dx = \int_c^d \int_a^b f(x, y) dx \, dy$

The Jacobian: Scaling Space

When we change variables (e.g., from Cartesian $(x, y)$ to Polar $(r, \theta)$), the “infinitesimal area” $dA$ changes. We use the Jacobian to account for this stretching.

• Cartesian: $dA = dx \, dy$
• Polar: $dA = r \, dr \, d\theta$

If you forget the $r$ in polar coordinates, your areas and volumes will be wrong!

from scipy import integrate
import numpy as np

# Let's calculate the volume of a hemisphere of radius 1
# Surface: z = sqrt(1 - x^2 - y^2)
# We will use SciPy to integrate over a circular region

def f(y, x):
  if x**2 + y**2 <= 1:
      return np.sqrt(1 - x**2 - y**2)
  return 0

# Integrate over the square [-1, 1] x [-1, 1]
# Note: we filter outside the circle in the function f
volume, error = integrate.dblquad(f, -1, 1, lambda x: -1, lambda x: 1)

print(f"Calculated Volume: {volume:.4f}")
print(f"Theoretical (2/3 * pi): { (2/3) * np.pi:.4f}")

Triple Integrals and Density

If $\rho(x, y, z)$ represents the density of an object at point $(x, y, z)$, the total mass of the object is the triple integral of the density over its volume $V$:

$M = \iiint_V \rho(x, y, z) dV$

Exercises