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The volume under a nonnegative continuous graph over a compact Jordan base is its integral

Statement

Let $D \subseteq R^{m}$ be compact and Jordan measurable and let $g : D \to [0, \infty)$ be continuous. Then $U_{g} : = {(u, t) : u \in D, 0 \leq t \leq g (u)}$ is compact and Jordan measurable, and

$cont (U_{g}) = \int_{D} g (u) d u .$

Facts & Assumptions

Given: The set $D$ , the nonnegative continuous function $g$ , and the constant integrand $H = 1$ on $U_{g}$ .

[F1]

If $D \subseteq R^{m}$ is compact and Jordan measurable and $α, β : D \to R$ are continuous with $α \leq β$ , then $K = {(u, t) : u \in D, α (u) \leq t \leq β (u)}$ is compact and Jordan measurable and every continuous $H : K \to R$ satisfies $\int_{K} H = \int_{D} (\int_{α (u)}^{β (u)} H (u, t) d t) d u$ (A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections).

Proof

technique · direct

1.1givenF1

Apply [F1] with lower graph $0$ , upper graph $g$ , and integrand $H = 1$ . This includes the empty base and the identically zero graph.

2.1step 1.1F1algebra∎

With $H = 1$ the left side of the formula in [F1] is $\int_{U_{g}} 1 = cont (U_{g})$ and the inner integral is $\int_{0}^{g (u)} 1 d t = g (u)$ , so that formula becomes $cont (U_{g}) = \int_{D} g$ .

Depends on

A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections

Dependency tree · 2 levels

Sources

Sigurd Angenent, Math 221 lecture notes, Chapter 8 §3.2