$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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proof

Volume of a set in $R^3$

Let $A\subseteq\mathbb{R}^2$ be a compact set with content, $f$ be a continuous and nonnegative function, and let $S:=\{(x, y, z)\in \mathbb{R}^3:(x, y)\in A, 0\leq z\leq f(x, y)\}$. Then $$\begin{align*} \mu(S)=\int_Af(x, y)d(x, y) \end{align*}$$

Concepts

Follows from Fubinis theorem.

Hypothesis

Coming soon.

Results

Coming soon

Proof

Coming soon

Volume of a set in $R^3$

Let $A\subseteq\mathbb{R}^2$ be a compact set with content, $f$ be a continuous and nonnegative function, and let $S:=\{(x, y, z)\in \mathbb{R}^3:(x, y)\in A, 0\leq z\leq f(x, y)\}$. Then $$\begin{align*} \mu(S)=\int_Af(x, y)d(x, y) \end{align*}$$

Concepts

Follows from Fubinis theorem.

Hypothesis

Coming soon.

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Volume of a set in $R^3$

Let $A\subseteq\mathbb{R}^2$ be a compact set with content, $f$ be a continuous and nonnegative function, and let $S:=\{(x, y, z)\in \mathbb{R}^3:(x, y)\in A, 0\leq z\leq f(x, y)\}$. Then $$\begin{align*} \mu(S)=\int_Af(x, y)d(x, y) \end{align*}$$

Concepts

Follows from Fubinis theorem.

Hypothesis

Coming soon.

Results

Coming soon

Proof

Coming soon

Volume of a set in $R^3$

Let $A\subseteq\mathbb{R}^2$ be a compact set with content, $f$ be a continuous and nonnegative function, and let $S:=\{(x, y, z)\in \mathbb{R}^3:(x, y)\in A, 0\leq z\leq f(x, y)\}$. Then $$\begin{align*} \mu(S)=\int_Af(x, y)d(x, y) \end{align*}$$

Concepts

Follows from Fubinis theorem.

Hypothesis

Coming soon.

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof