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

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the graphs of $f$ and $g$ intersect midway between $x=a$ and $x=b$, then
$\int_{a}^{b}[f(x)-g(x)] d x=0$

Key Concepts

Definite Integral

A definite integral represents the accumulation of a quantity, typically the net area between a function and the x-axis. In this context, it is used to compute the net difference between f(x) and g(x) over a specified interval, which involves adding positive areas (where f exceeds g) and subtracting negative areas (where g exceeds f).

Intersection of Functions

An intersection between two functions occurs at points where their values are equal. While an intersection might indicate a change in the relationship between the functions, the specific location of an intersection (such as midway in an interval) does not necessarily imply any symmetry in their behavior or the areas between them.

Symmetry in Integration

Symmetry in an integral refers to a balance in the function's behavior on an interval, such that areas on one side of a midpoint cancel out those on the other. The stated property would require symmetric behavior of the difference of functions about the midpoint, but a mere intersection at the midpoint does not guarantee this symmetry.

Net Area

The net area between two curves is the result of summing the areas where one function lies above the other and subtracting where it lies below. Even if f and g intersect at the midpoint, without additional symmetric properties in the functions, the areas on either side of the intersection might not be equal; therefore, the integral of their difference can be nonzero.

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