Area under Curve

Theorem

Let $f: \R \to \R$ be a real function which is defined and (Darboux) integrable on the closed interval $\closedint a b$.

Consider the graph of $f$ embedded in a Cartesian plane.

Let $\AA$ denote the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.

Then:

$\AA = \ds \int_a^b \map f x \rd x$

where $\ds \int_a^b \map f x \rd x$ denotes the (Darboux) definite integral of $f$ from $a$ to $b$.

Proof

Overview of Proof

Let $x \in \closedint a b$.

Let $\delta x$ be an arbitrarily small positive real number such that $x + \delta x \in \closedint a b$.

Consider the small strip between:

the $x$-axis

the vertical straight lines through $\tuple {x, 0}$ and $\tuple {x + \delta x, 0}$

the curve $\map f x$.

The area of this strip is approximated by $y \rdelta x$.

Hence we create a model of the geometric interpretation of the definite integral.

$\Box$

Lemma

Let $f : \closedint a b \to \R_{\mathop \ge 0}$ be non-negative and Darboux integrable over $\closedint a b$.

Let $G$ be the point set of $\tuple {x, y}$ such that $a \le x \le b$ and $0 \le y \le \map f x$.

Let $A$ be the area of $G$.

Then $A$ equals the Darboux integral of $f$ over $\closedint a b$.

$\Box$

Define $f^+$ and $f^-$ to be the positive and negative parts of $f$, respectively.

Let $A^+$ be the area under $\map {f^+} x$ on $\closedint a b$.

Let $A^-$ be the area under $\map {f^-} x$ on the same interval.

By Positive Part of Darboux Integrable Function is Integrable and its corollary for the negative part, $f^+$ and $f^-$ are Darboux integrable over $\closedint a b$.

By the Lemma, $A^+$ and $A^-$ are equal to the Darboux integral over their respective parts.

Therefore:

$\blacksquare$

Examples

Area under $y = \paren {x - 1} \paren {x - 2}$

The area between the $x$-axis and the curve $y = \paren {x - 1} \paren {x - 2}$ is $\dfrac 1 6$.

Area under $y = \sin x$ between $0$ and $\pi$

The area bounded by the curve $y = \sin x$ and the $x$-axis between $x = 0$ and $x = \pi$ is $2$.

Area between $x = y^2$ and $x^2 = 8 y$

Area under Curve/Examples/Between x=y^2 and x^2=8y

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Area, Volume and Centre of Gravity
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): area under a curve
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): area under a curve