Questions: Use your calculator to find the area shown in RED. The zero is not x=5 ! Round the zero to 4 decimals within your calculation. Total Area = [ ? ] Round your final answer to the nearest thousandth.

Solution Steps

To find the area between the curve \( f(x) = \sqrt{x^2 - 2x} - 4 \) and the x-axis, we need to determine the points where the curve intersects the x-axis. This occurs when \( f(x) = 0 \).

\[ \sqrt{x^2 - 2x} - 4 = 0 \] \[ \sqrt{x^2 - 2x} = 4 \] \[ x^2 - 2x = 16 \] \[ x^2 - 2x - 16 = 0 \]

Solving this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ x = \frac{2 \pm \sqrt{4 + 64}}{2} \] \[ x = \frac{2 \pm \sqrt{68}}{2} \] \[ x = \frac{2 \pm 2\sqrt{17}}{2} \] \[ x = 1 \pm \sqrt{17} \]

So, the intersection points are \( x = 1 + \sqrt{17} \) and \( x = 1 - \sqrt{17} \).

The area under the curve from \( x = 1 - \sqrt{17} \) to \( x = 1 + \sqrt{17} \) can be found by integrating \( f(x) \) over this interval.

\[ \text{Area} = \int_{1 - \sqrt{17}}^{1 + \sqrt{17}} \left( \sqrt{x^2 - 2x} - 4 \right) \, dx \]

To evaluate the integral, we can use numerical methods or a calculator.

Using a calculator to evaluate the integral:

\[ \int_{1 - \sqrt{17}}^{1 + \sqrt{17}} \left( \sqrt{x^2 - 2x} - 4 \right) \, dx \approx -32.000 \]

Final Answer