$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ expresses the area between the curves y = f(x) and y = g(x) from $x = a \operatorname { to } x = b$ only if:
A) f(x) is greater than or equal to g(x) for all x between a and b , and neither f(x) nor g(x) crosses the x-axis .
B) f(x) is greater than or equal to g(x) for all x between a and b .
C) f(x) and g(x) do not cross each other between x = a and x = b .
D) neither f(x) and g(x) cross the x-axis.
E) none of these

Question

Quizplus · Accepted Answer

The integral $\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ calculates the net area between the curves y = f(x) and y = g(x) from $x = a$ to $x = b$. For it to represent the area between the curves accurately, f(x) must be greater than or equal to g(x) throughout the interval [a, b]. This ensures that the integral captures the area between the curves without considering the direction of the area (above or below the x-axis), making the other conditions unnecessary for the calculation of the area between the curves.

∫ab(f(x)−g(x))dx\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x∫ab​(f(x)−g(x))dx

$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$