Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $f$ and $g$ are continuous on $[a, b]$ and either $f(x) \geq g(x)$ for all $x$ in $[a, b]$ or $f(x) \leq g(x)$ for all $x$ in $[a, b]$, then the area of the region bounded by the graphs of $f$ and $g$ and the vertical lines $x=a$ and $x=b$ is given by $\int_{a}^{b}|f(x)-g(x)| d x$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are continuous on $[a, b]$ and either $f(x) \geq g(x)$ for all $x$ in $[a, b]$ or $f(x) \leq g(x)$ for all $x$ in $[a, b]$, then the area of the region bounded by the graphs of $f$ and $g$ and the vertical lines $x=a$ and $x=b$ is given by $\int_{a}^{b}|f(x)-g(x)| d x$

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Key Concepts

Absolute Value in Integration

The absolute value ensures that the integrand is nonnegative, which is critical when calculating area since area cannot be negative. In cases where one function is consistently greater than or less than the other, the absolute value simply reduces to the difference between the function values.

Definite Integrals

Definite integrals provide a method to accumulate values over an interval, which in this context, compute the total area between two curves. The calculation involves integrating over the domain while ensuring that negative areas (from the ordering of the functions) are properly handled.

Continuous Functions

Continuous functions guarantee that there are no breaks or gaps in the graph, allowing the definite integral to accurately capture the overall area between the curves. This property is necessary for applying the fundamental theorem of calculus when working with integrals.

Area Between Graphs

The area between two curves over an interval is found by taking the integral of the absolute value of the difference between the function values. When one function is always above (or below) the other, the area is given simply by the integral of the difference, making this computation straightforward.

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