Double Integral Over Elementary Regions

Y-Simple

We want to integrate a real-valued function over a region $ϕ_{1} (x) \leq y \leq ϕ_{2} (x)$ and $x \in [a, b]$

\int \int_{R} f (x, y) d A = \int_{a}^{b} \int_{ϕ_{1} (x)}^{ϕ_{2} (x)} f (x, y) d y d x

if $D \subset R^{2}$ is y-simple then the intersection of any vertical line with D is either empty or a line segment

We want to integrate a real-valued function over a region $ψ_{1} (y) \leq x \leq ψ_{2} (y)$ and $y \in [d, c]$

\int \int_{R} f (x, y) d A = \int_{a}^{b} \int_{ψ_{1} (y)}^{ψ_{2} (y)} f (x, y) d x d y

if $D \subset R^{2}$ is y-simple then the intersection of any horizonal line with D is either empty or a line segment

A region that can be described as either Y-Simple or X-Simple

for example a unit-circle:

Y-simple

\begin{matrix} x \in [- 1, 1] \\ ϕ_{2} (x) = \sqrt{1 - x^{2}} \leq y \leq - \sqrt{1 - x^{2}} = ϕ_{1} (x) \end{matrix}

X-simple

\begin{matrix} y \in [- 1, 1] \\ ψ_{2} (y) = \sqrt{1 - y^{2}} \leq y \leq - \sqrt{1 - y^{2}} = ψ_{1} (y) \end{matrix}

The area of a region can be computed as:

\int_{α}^{β} \int_{θ_{1} (u)}^{θ_{2} (u)} 1 d v d u

you can think of this as a Riemann sum where $f (x, y) = I_{(x, y) \in D} (x, y)$

V = \int_{x_{min}}^{x_{max}} \int_{y_{min} (x)}^{y_{max} (x)} [f_{t o p} (x, y) - g_{b o t t o m} (x, y)] d y d x

$y = ϕ (x)$ rotation around x-axis

V = \int_{x_{m i n}}^{x_{m a x}} \int_{ϕ (x)}^{ϕ (x)} (θ of rotation) y d y d x

$x = ϕ (y)$ rotation around y-axis

V = \int_{y_{m i n}}^{y_{m a x}} \int_{ϕ (y)}^{ϕ (y)} (θ of rotation) x d x d y

$z = f (x, y)$ rotated around z-axis

V = 2 π \int_{x_{min}}^{x_{max}} \int_{y_{min} (x)}^{y_{max} (x)} | | (x, y) | | \cdot z d y d x

$z = f (x, y)$ rotated around x-axis

V = 2 π \int_{x_{min}}^{x_{max}} \int_{y_{min} (x)}^{y_{max} (x)} y \cdot z d y d x