The integral \(\int_{0}^{8} \sqrt{64 - x^2} \, dx\) can be interpreted geometrically and solved using the same principles as before.

### Step 1: Recognize the Geometry
The integrand \(\sqrt{64 - x^2}\) represents the upper half of a circle with the equation \(y = \sqrt{64 - x^2}\). This is the equation of a semicircle with a radius of \(8\) (since \(64 = 8^2\)) centered at the origin \((0, 0)\).

### Step 2: Sketch the Graph
The graph of \(y = \sqrt{64 - x^2}\) is a semicircle of radius \(8\). The integral from \(0\) to \(8\) represents the area under this semicircle from \(x = 0\) to \(x = 8\).

### Step 3: Calculate the Area
The integral \(\int_{0}^{8} \sqrt{64 - x^2} \, dx\) represents a quarter of the area of the full circle (since it's from \(x = 0\) to \(x = 8\)).

- The area of a full circle with radius \(8\) is given by \(A = \pi r^2 = \pi (8^2) = 64\pi\).
- The area of the upper half (a semicircle) is \(\frac{1}{2} \times 64\pi = 32\pi\).
- The area under the curve from \(x = 0\) to \(x = 8\) (a quarter-circle) is \(\frac{1}{4} \times 64\pi = 16\pi\).

### Step 4: Final Answer
Thus, the value of the integral \(\int_{0}^{8} \sqrt{64 - x^2} \, dx\) is:

\[
\int_{0}^{8} \sqrt{64 - x^2} \, dx = 16\pi
\]

Would you like further details or have any other questions? Here are five related questions you might find interesting:

1. What would the integral be if the limits were from \(-8\) to \(8\)?
2. Can you calculate the integral using trigonometric substitution instead of geometry?
3. How would the integral change if the radius of the circle was doubled?
4. What is the significance of the integral \(\int_{0}^{r} \sqrt{r^2 - x^2} \, dx\) for any radius \(r\)?
5. How can you compute \(\int_{0}^{8} \sqrt{64 - x^2} \, dx\) if you were given only basic integral tables?

**Tip:** When dealing with integrals of the form \(\int \sqrt{a^2 - x^2} \, dx\), it's often useful to think of them in terms of the geometry of circles. This approach can simplify calculations significantly.

Learn how to calculate the integral ∫(0 to 8) sqrt(64 - x^2) dx with a detailed step-by-step solution. This problem involves the geometric interpretation of integrals, focusing on the area under a semicircle. Suitable for advanced high school students and anyone interested in integral calculus.

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