Calculate Negative Area of f(x) = x² - 16: Below X-Axis Integration

Find the negative area of the function

$f(x)=x^2-16$

Video Solution

Solution Steps

00:00 Find the negative domain of the function

00:03 Notice the coefficient of X squared is positive, so the function is happy (opens upward)

00:08 The negative domain is actually below the X-axis

00:13 Therefore, substitute Y=0 to find the intersection points with the X-axis

00:18 Isolate X

00:23 Take the square root

00:31 When taking a square root, there are 2 solutions (positive and negative)

00:37 These are the intersection points with the X-axis

00:43 Let's mark the intersection points with the X-axis

00:54 The positive domain is above the X-axis

00:59 The negative domain is below the X-axis

01:05 Let's find the domain where the function is negative

01:11 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we find where the function $f(x) = x^2 - 16$ is negative.

Step 1: Set $f(x) = 0$ to find where the function has zero values, solving $x^2 - 16 = 0$ .
Step 2: Factor the equation as $(x - 4)(x + 4) = 0$ .
Step 3: Find the roots: $x = 4$ and $x = -4$ .
Step 4: Test intervals: $x < -4$ , $-4 < x < 4$ , and $x > 4$ . Inside the interval $-4 < x < 4$ , $x^2 - 16$ is negative because the square of any number less than 4 but greater than -4 will be less than 16.

Therefore, the function $f(x)$ is negative on the interval $-4 < x < 4$ .