Total area

Definition

Total area is the absolute value of the sum of the areas between a function and the x-axis over a given interval. It accounts for all regions, both above and below the x-axis, by considering their absolute values.

5 Must Know Facts For Your Next Test

1. Total area differs from net area because it does not cancel out regions below the x-axis; instead, it sums their absolute values.
2. To calculate total area using integration, you often need to split the integral at points where the function crosses the x-axis.
3. The total area can be found by integrating $|f(x)|$ over the interval if $f(x)$ is continuous.
4. When dealing with multiple intervals where $f(x)$ changes sign, each segment must be integrated separately to ensure correct calculation.
5. Total area is particularly useful in applications where negative areas are not meaningful, such as in physics for calculating work done.

Review Questions

• What is the difference between total area and net area?
• How do you handle integration when a function crosses the x-axis within an interval?
• Why might you integrate $|f(x)|$ instead of $f(x)$ when calculating total area?

Related terms

Definite Integral: $\int_{a}^{b} f(x) \, dx$ represents the signed area under $f(x)$ from $x=a$ to $x=b$.

Net Area: The algebraic sum of areas between a function and the x-axis over a given interval, taking into account positive and negative regions.

$|f(x)|$: $|f(x)|$ denotes the absolute value of $f(x)$, used to ensure all contributions to total area are positive.