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

Find the negative area of the function

f(x)=x2−16 f(x)=x^2-16

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
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 written solution

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1

Understand the problem

Find the negative area of the function

f(x)=x2−16 f(x)=x^2-16

2

Step-by-step solution

To solve this problem, we find where the function f(x)=x2−16 f(x) = x^2 - 16 is negative.

  • Step 1: Set f(x)=0 f(x) = 0 to find where the function has zero values, solving x2−16=0 x^2 - 16 = 0 .
  • Step 2: Factor the equation as (x−4)(x+4)=0 (x - 4)(x + 4) = 0 .
  • Step 3: Find the roots: x=4 x = 4 and x=−4 x = -4 .
  • Step 4: Test intervals: x<−4 x < -4 , −4<x<4 -4 < x < 4 , and x>4 x > 4 . Inside the interval −4<x<4 -4 < x < 4 , x2−16 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) f(x) is negative on the interval −4<x<4 -4 < x < 4 .

3

Final Answer

−4<x<4 -4 < x < 4

Practice Quiz

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\( y=-2x^2-3 \)

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