Non-Rectangular Bounds
If the region of integration isn't a rectangle, the limits of the inner integral will depend on the variable of the outer integral.
Example: \(\int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) dy dx\).
Worked Examples
Example 1: The Triangle
Evaluate \(\iint_D (x+y) dA\) where \(D\) is the triangle bounded by \(x=0, y=0, y=1-x\).
- Bounds: \(x\) from 0 to 1. For each \(x\), \(y\) goes from 0 to \(1-x\).
- Integral: \(\int_0^1 \int_0^{1-x} (x+y) dy dx\).
- Inner: \([xy + y^2/2]_0^{1-x} = x(1-x) + (1-x)^2/2 = x - x^2 + (1 - 2x + x^2)/2 = 0.5 - 0.5x^2\).
- Outer: \(\int_0^1 (0.5 - 0.5x^2) dx = [0.5x - x^3/6]_0^1 = 0.5 - 1/6 = 1/3\).
- Result: 1/3.
The Bridge to Quantum Mechanics
Particles are rarely trapped in perfect rectangles. An electron in a chemical bond exists in a region defined by the overlap of two atoms. This "General Region" requires us to set up integrals with variable bounds. Mastering this is the only way to calculate the Bond Energy between atoms, which is the fundamental force that holds all matter together.