List of basic formulas related to double integrals:
- Definition of a double integral:
[
\iint_D f(x, y) \, dA = \int_a^b \int_c^df(x, y) \, dy \, dx
]
where (D) is the domain of integration and (dA) is the area element. - Moving to polar coordinates:
[
\iint_D f(x, y) \, dA = \iint_D f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta
]
where (r) is the radius, and (\theta) is the angle. - Properties of a double integral:
- Linearity:
[
\iint_D (af(x, y) + bg(x, y)) \, dA = a \iint_D f(x, y) \, dA + b \iint_D g(x, y) \, dA
] - Non-negativity:
[
\text{If } f(x, y) \geq 0 \text{ on } D, \text{ then } \iint_D f(x, y) \, dA \geq 0
]
- Foobini formula:
If (f(x, y)) is continuous on a rectangular domain (D), then:
[
\iint_D f(x, y) \, dA = \int_a^b \left( \int_c^df(x, y ) \, to \right) dx = \int_c^d \left( \int_a^bf(x, y) \, dx \right)
to - Double integral over the domain given by the inequalities:
If the domain (D) is given by the inequalities (g_1(x) \leq y \leq g_2(x)) and (a \leq x \leq b), then:
[
\iint_D f(x, y) \, dA = \int_a^b \left( \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \right) dx
]