Let $H$ be the hemisphere $x^2 + y^2 + z^2 = 45$ , $z \geq 0$ , and suppose $f$ is a continuous function with $f(5, 2, 4) = 5$ , $f(5, -2, 4) = 7$ , $f(-5, 2, 4) = 8$ , and $f(-5, -2, 4) = 11$ . By dividing $H$ into four patches, estimate the value below. (Round your answer to the nearest whole number.)

$\iint_H f(x, y, z) \, dS$

Question :

Let $h$ be the hemisphere $x^2 + y^2 + z^2 = 45$ , $z \geq 0$ , and suppose $f$ is a continuous function with $f(5, 2, 4) = 5$ , $f(5, -2, 4) = 7$ , $f(-5, 2, 4) = 8$ , and $f(-5, -2, 4) = 11$ . by dividing $h$ into four patches, estimate the value below. (round your answer to the nearest whole number.)

$\iint_h f(x, y, z) \, ds$

$Let h be the hemisphere x^2 + y^2 + z^2 = 45, z \geq 0, and suppose f is | Doubtlet.com$

Solution:

Neetesh Kumar | November 10, 2024

Calculus Homework Help

This is the solution to Math 1D
Assignment: 16.7 Question Number 1
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Step-by-step solution:

To estimate the surface integral $\iint_H f(x, y, z) \, dS$ by dividing $H$ into four patches, we will multiply the average value of $f$ over each patch by the approximate area of each patch.

Step 1: Approximate the Area of $H$

Since $H$ is the upper hemisphere of a sphere with radius $\sqrt{45}$ , the surface area of $H$ is half the surface area of a full sphere with radius $\sqrt{45}$ .

The surface area $A$ of a sphere of radius $R$ is given by: $A = 4\pi R^2$

For $H$ , with $R = \sqrt{45}$ : $A_H = 2\pi (\sqrt{45})^2 = 2\pi \cdot 45 = 90\pi$

Step 2: Divide $H$ into Four Equal Patches

Since we are dividing $H$ into four patches, each patch will have an approximate area of: $\frac{A_H}{4} = \frac{90\pi}{4} = 22.5\pi$

Step 3: Estimate the Integral Using the Values of $f$ at Each Patch Center

We approximate the integral by taking the sum of $f(x, y, z)$ values at the given points and multiplying by the area of each patch: $\iint_H f(x, y, z) \, dS \approx \left( f(5, 2, 4) + f(5, -2, 4) + f(-5, 2, 4) + f(-5, -2, 4) \right) \cdot \frac{A_H}{4}$

Substituting the given values of $f$ : $\iint_H f(x, y, z) \, dS \approx (5 + 7 + 8 + 11) \cdot 22.5\pi$ $= 31 \cdot 22.5\pi$ $= 697.5\pi$

Step 4: Approximate the Value

Using $\pi \approx 3.1416$ : $697.5\pi \approx 697.5 \times 3.1416 = 2191.55$

Rounding to the nearest whole number: $\iint_H f(x, y, z) \, dS \approx 2192$

Final Answer:

$\iint_H f(x, y, z) \, dS \approx \boxed{2192}$

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Calculus Homework Help

Step-by-step solution:

Step 1: Approximate the Area of HHH

Step 2: Divide HHH into Four Equal Patches