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

00:05 Let's find where the function is negative, below the X-axis.

00:09 The X squared coefficient is positive. So the curve is happy and opens upward.

00:15 Remember, the negative domain is the part under the X-axis.

00:20 To find X-intercepts, set Y equal to zero.

00:24 Now, let's isolate X in the equation.

00:28 Next step, take the square root. Remember, it gives both a positive and negative solution.

00:36 These solutions are where the graph crosses the X-axis.

00:42 Let's put these crossing points on the graph.

00:48 We can see the positive domain is above the X-axis.

00:59 And remember, the negative domain is what lies below the X-axis.

01:04 Let's focus on finding the negative domain, where the graph dips below.

01:10 And that's how we find where the function is negative!

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 $-4 < x < 4$ .