Multiple Integrals

🔢 Multiple Integrals

Multiple integrals allow us to calculate areas, volumes, masses, and other quantities over regions of two or more dimensions.

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🟢 1. Double Integrals ($\iint$)

The double integral of a function $f(x, y)$ over a region $R$ in the $xy$-plane represents the volume of the solid below the surface $z = f(x, y)$ and above $R$. $\iint_R f(x, y) dA$

Fubini’s Theorem

For a rectangular region $R = [a, b] \times [c, d]$, we can evaluate the double integral as an iterated integral: $\iint_R f(x, y) dA = \int_a^b \int_c^d f(x, y) dy dx = \int_c^d \int_a^b f(x, y) dx dy$

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🟡 2. Double Integrals in Polar Coordinates

When the region $R$ is circular or disk-like, polar coordinates are often simpler.

• $x = r \cos(\theta)$
• $y = r \sin(\theta)$
• Area element: $dA = r dr d\theta$

$\iint_R f(x, y) dA = \iint_{R'} f(r \cos\theta, r \sin\theta) \cdot r dr d\theta$

Caution

Don’t forget the extra $r$ when switching to polar!

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🔴 3. Triple Integrals ($\iiint$)

Triple integrals calculate the sum of values of $f(x, y, z)$ over a 3D region $E$. $\iiint_E f(x, y, z) dV$

Coordinate Systems for 3D

• Cylindrical ($r, \theta, z$): Used for solids with axis symmetry. $dV = r dz dr d\theta$.
• Spherical ($\rho, \phi, \theta$): Used for spheres or cones. $dV = \rho^2 \sin\phi d\rho d\phi d\theta$.

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🎯 4. Change of Variables & The Jacobian

When changing coordinate systems (from $u, v$ to $x, y$), we scale the integral by the Jacobian: $\iint_R f(x, y) dx dy = \iint_{R'} f(x(u, v), y(u, v)) \cdot \left| \frac{\partial(x, y)}{\partial(u, v)} \right| du dv$

The Jacobian determinant $\left| \frac{\partial(x, y)}{\partial(u, v)} \right|$ represents the “area-scaling factor” of the transformation.

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💡 Practical Example: Mass Calculation (Numerical Integration)

If $f(x, y)$ represents the density of a thin plate, the double integral over the region gives the total mass.

import numpy as np

def calculate_mass_grid(density_func, x_range, y_range, n_pts):
    x = np.linspace(x_range[0], x_range[1], n_pts)
    y = np.linspace(y_range[0], y_range[1], n_pts)
    dx = x[1] - x[0]
    dy = y[1] - y[0]
    
    X, Y = np.meshgrid(x, y)
    Z = density_func(X, Y)
    
    # 2D approximation (simple sum)
    mass = np.sum(Z) * dx * dy
    return mass

# Example: Density f(x, y) = x*y on [0, 1]x[0, 1]
# Analytical: 1/2 * 1/2 = 0.25
mass = calculate_mass_grid(lambda x, y: x*y, (0, 1), (0, 1), 500)
print(f"Approximated Mass: {mass}")

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🚀 Key Takeaways

• Double integrals find volume under a surface.
• Iterated integrals let us solve one variable at a time.
• Choosing the right coordinate system (Polar/Spherical) makes integration much easier.