Total area

Definition

Total area is the sum of the absolute values of the areas between a curve and the x-axis over a given interval. It accounts for both positive and negative regions by considering their absolute magnitudes.

5 Must Know Facts For Your Next Test

1. Total area is calculated using the definite integral with absolute values: $\int_a^b |f(x)| dx$.
2. Unlike net area, total area considers all regions as positive, regardless of whether they lie above or below the x-axis.
3. When calculating total area, partitioning at points where $f(x)=0$ can help separate different regions of integration.
4. Total area is useful in applications requiring a measure of space covered without regard to direction.
5. Graphical interpretation involves shading all areas between the curve and x-axis positively.

Review Questions

• What is the main difference between net area and total area?
• How do you modify the definite integral to calculate total area?
• Why might it be necessary to find points where $f(x)=0$ when computing total area?

Related terms

Definite Integral:

The calculation of the net signed area under a curve between two bounds, represented as $\int_a^b f(x) dx$.

Net Area: The difference between areas above and below the x-axis for a given function over an interval.

$|f(x)|$: $|f(x)|$, or absolute value function, converts all output values to non-negative equivalents, essential for computing total area.