Q. 10 - Calculus

Question

Use the definition of absolute value to explain why the absolute area ${\int }_a^b\left|f\left(x\right)\right| dx$ counts area positively regardless of whether it is above or below the x-axis.

Step-by-Step Solution

Answer

The absolute area of ${\int }_a^b\left|f\left(x\right)\right| dx$ counts to be positive regardless of whether it is above or below the x-axis.

Step 1. Given information.

Consider the given question,

${\int }_a^b\left|f\left(x\right)\right| dx$

Step 2. Find the absolute area.

Absolute value means to remove any negative sign in front of a number making the number positive.

So, the expression $\left|f\left(x\right)\right|$ counts positive values.

Therefore, the absolute area of ${\int }_a^b \left|f\left(x\right)\right| dx$ counts to be positive regardless of whether it is above or below the x-axis.