Area under Curve
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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
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 $y \delta x$.
Hence we create a model of the geometric interpretation of the definite integral.
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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