let H be the hemisphere 
x2 + y2 + z2 = 45, z ≥ 0,
 and suppose f is a continuous function with 
f(4, 2, 5) = 8,
 
f(4, −2, 5) = 9,
 
f(−4, 2, 5) = 10,
 and 
f(−4, −2, 5) = 12.
 By dividing H into four patches, estimate the value below. (Round your answer to the nearest whole number.)

To estimate the value of the integral of \( f \) over the hemisphere \( H \), we'll approximate this by dividing the hemisphere into four patches and evaluating the integral at the center of each patch.

Given:

- The hemisphere \( H \) is defined by \( x^2 + y^2 + z^2 = 45 \) and \( z \geq 0 \).
- The function \( f \) is continuous and the values of \( f \) at specific points are given as follows:
  - \( f(4, 2, 5) = 8 \)
  - \( f(4, -2, 5) = 9 \)
  - \( f(-4, 2, 5) = 10 \)
  - \( f(-4, -2, 5) = 12 \)

### Step 1: Describe the Hemisphere

The hemisphere is the upper half of the sphere \( x^2 + y^2 + z^2 = 45 \) above the \( xy \)-plane. We can divide this hemisphere into four regions or patches. The hemisphere will be divided along the planes \( x = 0 \) and \( y = 0 \), creating four quadrants in the \( xy \)-plane.

### Step 2: Divide into Four Patches

The four patches correspond to the four quadrants in the \( xy \)-plane:

1. \( x > 0, y > 0 \) with the center at \( (4, 2, 5) \)
2. \( x > 0, y < 0 \) with the center at \( (4, -2, 5) \)
3. \( x < 0, y > 0 \) with the center at \( (-4, 2, 5) \)
4. \( x < 0, y < 0 \) with the center at \( (-4, -2, 5) \)

### Step 3: Estimate the Integral

The integral over the hemisphere \( H \) can be approximated by summing the function values at these four points and then multiplying by the approximate area of each patch.

Since the hemisphere is symmetric, each patch should cover approximately the same area.

- Area of the hemisphere \( H \) is \( 2\pi R^2 \) where \( R = \sqrt{45} = 3\sqrt{5} \).
- The area of each patch is \( \frac{2\pi (3\sqrt{5})^2}{4} = \frac{2\pi \cdot 45}{4} = 22.5\pi \).

The estimated value of the integral is given by:

\[
\text{Integral over } H \approx \frac{\text{Area of each patch}}{4} \times \left( f(4, 2, 5) + f(4, -2, 5) + f(-4, 2, 5) + f(-4, -2, 5) \right)
\]

Substituting the values:

\[
\text{Integral over } H \approx \frac{1}{4} \times 22.5\pi \times (8 + 9 + 10 + 12)
\]

\[
\text{Integral over } H \approx \frac{1}{4} \times 22.5\pi \times 39
\]

\[
\text{Integral over } H \approx 22.5\pi \times 9.75
\]

\[
\text{Integral over } H \approx 219.375\pi
\]

Finally, rounding to the nearest whole number:

\[
\text{Integral over } H \approx 689
\]

### Final Answer
The estimated value is **689**.

Would you like more details or have any questions? Here are five questions to expand on this:

1. How would the answer change if the function \( f \) were not continuous?
2. What if the radius of the hemisphere were different, how would that affect the estimate?
3. Can you explain how symmetry helps simplify the integral approximation in this problem?
4. How would you perform this integral exactly instead of using the approximation method?
5. What changes if we consider a quarter-sphere instead of a hemisphere?

**Tip:** When estimating integrals, especially on symmetric regions, symmetry can significantly simplify the computation, reducing complex integrals into simpler summations.

Learn how to estimate the integral of a continuous function over a hemisphere divided into four patches. This problem involves surface integrals, spherical coordinates, and integral approximation methods. Explore step-by-step solutions and related mathematical concepts.

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