﻿Use geometry 
(
not Riemann sums
)
 ﻿to evaluate the definite integral. Sketch the graph of the integrand, show the region in question, and interpret your
result.
∫
0
8
6
4
-
x
2
d
x
Choose the correct graph below.
A
.
B
.
C
.
D
.

To evaluate the definite integral \(\int_{0}^{8} (64 - x^2) \, dx\) using geometry, we need to interpret the integral as the area under the curve \(y = 64 - x^2\) from \(x = 0\) to \(x = 8\). This function represents a downward-opening parabola with its vertex at \((0, 64)\). The graph will intersect the \(x\)-axis where \(64 - x^2 = 0\), i.e., \(x = \pm 8\).

### Step-by-Step Solution:

1. **Identify the Shape of the Graph:**
   The function \(y = 64 - x^2\) is a downward-facing parabola. To sketch the graph, we note:
   - The vertex of the parabola is at \((0, 64)\).
   - The parabola intersects the \(x\)-axis at \(x = 8\) and \(x = -8\).
   
2. **Determine the Region in Question:**
   We need to find the area under the curve from \(x = 0\) to \(x = 8\). The region of interest is from \(x = 0\) to \(x = 8\), which is a symmetric portion of the parabola with respect to the \(y\)-axis.

3. **Geometric Interpretation:**
   The area under the parabola from \(x = 0\) to \(x = 8\) forms a segment of a larger semicircle. Since the equation is quadratic, it corresponds to half of a circle.

   The standard form of the equation of a semicircle with radius \(r\) is \(y = \sqrt{r^2 - x^2}\), which closely matches our function if we rewrite \(64 - x^2 = (8^2 - x^2)\). Thus, this represents the top half of a circle centered at \((0, 0)\) with a radius of 8.

4. **Calculate the Area Using Geometry:**
   The total area of the circle with radius 8 is \(\pi r^2 = \pi \times 8^2 = 64\pi\). Since we are dealing with the top half of the circle, the area under the curve from \(-8\) to \(8\) is \(\frac{1}{2} \times 64\pi = 32\pi\).

   However, we only need the area from \(x = 0\) to \(x = 8\), which is half of the semicircle:
   \[
   \text{Area} = \frac{1}{2} \times 32\pi = 16\pi.
   \]

### Conclusion:
The area under the curve from \(x = 0\) to \(x = 8\) is \(16\pi\). This is the value of the definite integral \(\int_{0}^{8} (64 - x^2) \, dx = 16\pi\).

### Graph Selection:
To select the correct graph:
- The graph should display a downward-facing parabola.
- The parabola should intersect the \(x\)-axis at \(x = -8\) and \(x = 8\), and have a vertex at \((0, 64)\).
- The shaded region should be from \(x = 0\) to \(x = 8\).

Please provide the images of options A, B, C, and D so I can help identify the correct graph.

Learn how to evaluate the definite integral ∫(64 - x^2) dx from 0 to 8 using geometric interpretation. Understand the concept of integrating polynomials and interpreting the area under a curve as the top half of a circle. Suitable for advanced high school students.

Math Problem Statement

Use geometry ( not Riemann sums ) to evaluate the definite integral. Sketch the graph of the integrand, show the region in question, and interpret your result. ∫ 0 8 6 4

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