Double Integral
The double integral of \(f\) over the rectangle \(R\) is
\[\iint*R f(x,y)dA = \lim\_{m,n \rightarrow \infty} \sum\_{i=1}^m \sum\_{j=1}^n f(x*{ij}^\*, y\_{ij}^\*) \Delta A\]If \(f(x,y) \geq 0\), then the volumne \(V\) of the solid that lies above the rectangle \(R\) and below the surface \(z=f(x,y)\) is
\[V = \iint_R f(x,y)dA\]\[\iint*R f(x,y)dA \sim \sum*{i=1}^m \sum\_{j=1}^n f(\bar{x_i}, \bar{y_j}) \Delta A\]Midpoint Rule for Double Integrals
where \(\bar{x_i}\) is the midpoint of \([x_{i-1},x_i]\) and \(\bar{y_j}\) is the midpoint of \([y_{j-1},y_j]\).
Fubini’s Theorem
If \(f\) is continuous on the rectangle \(R=\{ (x,y) \vert a \leq x \leq b, c \leq y \leq d \}\), then
\[\iint_R f(x,y)dA = \int_a^b \int_c^d f(x,y)dydx = \int_c^d \int_a^bf(x,y)dxdy\]Average Value
The average value of a function \(f\) of two variables defined on a rectangle \(R\) is
\[f\_{ave} = \frac{1}{A(R)} \iint_R f(x,y)dA\]where \(A(R)\) is the area of \(R\).
References
[1] Stewart Calculus, 8th edition