Let f and g be continuous function on a ≤x ≤b and set...

Let $f$ and $g$ be continuous function on $a \leq x \leq b$ and set $p(x)=\max \{f(x), g(x)\}$ and $q(x)=\min \{f(x), g(x)\}$, then the area bounded by the curves $y=p(x), y=q(x)$ and the ordinates $x=a$ and $x=b$ is given by (a) $\int_{4}^{b}|f(x)-g(x)| d x$ (b) $\int_{a}^{b}|p(x)-q(x)| d x$ (c) $\left.\int_{a}^{b} \mid f(x)-g(x)\right\} d x$ (d) $\int_{a}^{b}\{p(x)-q(x)\} d x$

Let $f$ and $g$ be continuous function on $a \leq x \leq b$ and set $p(x)=\max \{f(x), g(x)\}$ and $q(x)=\min \{f(x), g(x)\}$,
then the area bounded by the curves $y=p(x), y=q(x)$
and the ordinates $x=a$ and $x=b$ is given by
(a) $\int_{4}^{b}|f(x)-g(x)| d x$
(b) $\int_{a}^{b}|p(x)-q(x)| d x$
(c) $\left.\int_{a}^{b} \mid f(x)-g(x)\right\} d x$
(d) $\int_{a}^{b}\{p(x)-q(x)\} d x$

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Key Concepts

Continuous Functions

The property of continuity guarantees that the functions do not have any breaks or abrupt jumps over the interval. This ensures that both the area between the curves and the resulting integrals are well-defined and can be accurately computed.

Area Between Curves

This concept involves finding the area enclosed between two curves over a specified interval. The general method requires determining the vertical distance between the curves at each x-value and integrating this distance over the interval to compute the total area.

Absolute Value Integration

In computing areas, taking the absolute value of the difference between two functions ensures that the integrals represent actual areas, as distances are inherently non-negative. This is important when one function may dip below the other over some parts of the interval.

Maximum and Minimum Functions

The max and min functions select the higher and lower of two function values at any given point, respectively. They succinctly identify the bounds of the region between the curves, which is crucial when setting up integrals to calculate the area between them.

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