https://file.aiquickdraw.com/mathbot/file/1731383141461uo6ly44h.jpg

Let's break down each question in the image to find solutions.

### Problem 1
The question is:
> Find the area of the surface.  
> The part of the surface \( 2y + 4z - x^2 = 5 \) that lies above the triangle with vertices \((0, 0)\), \((2, 0)\), and \((2, 4)\).

This is a surface area problem where we need to find the surface area above a given region. Here’s how we can approach it:
1. Rewrite the surface equation \(2y + 4z = x^2 + 5\) to isolate \(z\):  
   \[
   z = \frac{x^2 - 2y + 5}{4}
   \]
2. Set up the surface area integral over the triangular region defined by the vertices.
3. Compute partial derivatives of \(z\) with respect to \(x\) and \(y\), and then use the surface area formula.

### Problem 2
The question is:
> Find the area of the surface.  
> The part of the hyperbolic paraboloid \(z = y^2 - x^2\) that lies between the cylinders \(x^2 + y^2 = 9\) and \(x^2 + y^2 = 16\).

Here’s the approach for this:
1. The surface \(z = y^2 - x^2\) is a hyperbolic paraboloid.
2. We need to find the surface area between the regions bounded by the two cylinders. This means we’ll set up an integral in polar coordinates with \(r\) ranging from \(3\) (from \(x^2 + y^2 = 9\)) to \(4\) (from \(x^2 + y^2 = 16\)).
3. Convert the expression to polar coordinates and set up the surface area integral.

### Problem 3
The question is:
> Find the area of the surface.  
> The part of the sphere \(x^2 + y^2 + z^2 = 9\) that lies above the plane \(z = 1\).

For this problem:
1. Recognize that \(x^2 + y^2 + z^2 = 9\) is a sphere with radius 3.
2. The plane \(z = 1\) intersects the sphere, forming a circular cap on the sphere.
3. Use spherical coordinates or set up a surface area integral for a spherical cap.

Would you like the full solutions and integrals set up for each problem?

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### Additional Questions for Further Understanding:
1. How would you set up polar coordinates for surface area calculations?
2. What is the process for finding partial derivatives in surface area problems?
3. How does the radius of a spherical cap affect its surface area?
4. Can you explain how the hyperbolic paraboloid differs in shape from a sphere?
5. What is the relationship between a surface's equation and its partial derivatives?

#### Tip:
When dealing with complex surfaces, breaking the problem down into manageable steps like isolating \(z\), determining the region of integration, and choosing the correct coordinate system is essential for efficient problem-solving.

Find the area of the surface for various surfaces within specified regions.

Solve surface area problems involving a plane above a triangular region, a hyperbolic paraboloid between cylindrical boundaries, and a spherical cap above a plane. Learn to set up surface integrals and use polar coordinates in multivariable calculus.

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