Area Under the Curve

Definite integral can yield out negative and positive output (accumulation of change).

Since the area is always positives

• y=f(x) is nonnegative (the curve is above the x axis) over a closed interval [a,b], then the area under the curve of y=f(x) from a to b is the definite integral of f from a to b.
• y=f(x) is negative (the curve is under the x axis) over a closed interval [a,b], then the area under the curve of y=f(x) from a to b is the opposite of the definite integral of f from a to b.

Therefore, $\int _a ^b |f(x)|dx$ is the most consistent when finding the area.

On the other hand, we can break the integral down to smaller integral on the x intercept (c) that makes the function to change its sign. [or the points where a piece-wise function have break into two different pieces)

$$ \int_a^bf(x)dx=\int_a^cf(x)dx+\int_c^bf(x)dx $$