Double integrals: ∬f(x,y)dA = ∫ₐᵇ∫ₓᵈ f(x,y)dydx (iterated). Volume under z=f(x,y) above region R. Polar: ∬f(r,θ)r·dr·dθ (Jacobian r). Fubini's theorem allows swapping order of integration.
How to Use
Set up the integral:
- f(x,y): Choose function
- Bounds: x∈[a,b], y∈[c,d]
- Grid: n×m subdivisions
Interpretation
If f(x,y)≥0: volume under the surface. If f=1: area of region R. If f=ρ(x,y): mass of lamina. Center of mass, moments of inertia all use double integrals.
Polar Coordinates
Use when region is circular. x=rcosθ, y=rsinθ, dA=r·dr·dθ. Circles, annuli, and sectors are natural in polar coordinates.
Step-by-Step Instructions
- 1Select a function.
- 2Set x bounds.
- 3Set y bounds.
- 4Choose grid density.
- 5View approximation.