Triple integrals in cylindrical coordinates are a game-changer for solving problems with circular symmetry. They transform complex 3D shapes into manageable calculations, making life easier when dealing with cylinders, cones, or spheres.

This method replaces x and y with r and θ, keeping z as is. The magic happens when you swap dV for r dr dθ dz, simplifying your integrals. It's like having a secret weapon for tackling tricky volume problems.

Integration Setup

Setting Up Triple Integrals and Integration Limits

• Triple integrals evaluate a function $f(x, y, z)$ over a three-dimensional region $E$
  • Denoted as $\iiint_E f(x, y, z) , dV$, where $dV$ represents the volume element
• Integration limits define the boundaries of the region $E$
  • Determined by the given problem or the shape of the region (rectangular box, cylinder, sphere)
• Order of integration specifies the sequence in which the variables are integrated
  • Common orders: $dz , dy , dx$, $dz , dx , dy$, $dy , dz , dx$, $dy , dx , dz$, $dx , dz , dy$, $dx , dy , dz$
  • Choice of order depends on the region and the integrand

Iterated Integrals and Evaluation

• Iterated integrals break down the triple integral into a sequence of single integrals
  • Evaluate the innermost integral first, treating other variables as constants
  • Substitute the result into the next integral and evaluate, repeating until all integrals are computed
• Example: $\int_a^b \int_c^d \int_e^f f(x, y, z) , dz , dy , dx = \int_a^b \int_c^d \left[ \int_e^f f(x, y, z) , dz \right] dy , dx$
  • Evaluate $\int_e^f f(x, y, z) , dz$ first, treating $x$ and $y$ as constants
  • Substitute the result into $\int_c^d \left[ \cdots \right] dy$ and evaluate, treating $x$ as constant
  • Finally, evaluate $\int_a^b \left[ \cdots \right] dx$

Cylindrical Coordinates

Cylindrical Volume Element and Change of Variables

• Cylindrical coordinates $(r, \theta, z)$ are useful for regions with cylindrical symmetry
  • $r$: distance from the z-axis (radius)
  • $\theta$: angle in the xy-plane (azimuth)
  • $z$: height along the z-axis
• Change of variables from Cartesian $(x, y, z)$ to cylindrical $(r, \theta, z)$:
  • $x = r \cos \theta$
  • $y = r \sin \theta$
  • $z = z$
• Cylindrical volume element: $dV = r , dr , d\theta , dz$
  • Derived using the Jacobian determinant: $\left| \frac{\partial(x, y, z)}{\partial(r, \theta, z)} \right| = r$

Cylindrical Shells and Integration

• Cylindrical shells are concentric hollow cylinders of infinitesimal thickness
  • Used to calculate volumes of solids of revolution
• Integration using cylindrical shells:
  • $V = \int_a^b 2\pi r h(r) , dr$, where $h(r)$ is the height of the shell at radius $r$
  • Radius $r$ varies from $a$ to $b$, and each shell has a thickness of $dr$
• Example: Find the volume of a cone with radius $R$ and height $H$
  • $h(r) = \frac{H}{R}(R - r)$, where $r$ varies from $0$ to $R$
  • $V = \int_0^R 2\pi r \frac{H}{R}(R - r) , dr = \frac{1}{3}\pi R^2 H$