Answer:
To find the surface area of the surface S, we need to find the area of the curved surface of the cylinder and the area of the plane surface.
First, let's find the equation of the cylinder. We can see that the sides of the cylinder are given by the equations y = 4x^2 and y = 8 - 4x^2. These are equations of parabolas with vertex at the origin and axis along the y-axis. The cylinder is formed by sweeping the region between the two parabolas along the y-axis. We can write the equation of the cylinder in terms of x and y as:
4x^2 ≤ y ≤ 8 - 4x^2
Next, let's find the intersection of the cylinder with the plane. We can substitute 3x + 4y + 12z = 7 into the equation of the cylinder:
4x^2 ≤ y ≤ 8 - 4x^2
3x + 4y + 12z = 7
Solving for z, we get:
z = (7 - 3x - 4y) / 12
Substituting this into the equation of the cylinder, we get:
4x^2 ≤ y ≤ 8 - 4x^2
z = (7 - 3x - 4y) / 12
This gives us the equation of the surface S.
To find the surface area of S, we need to integrate the square root of the sum of the squares of the partial derivatives of z with respect to x and y over the region of S. This is given by the surface area integral:
∫∫S √(1 + (dz/dx)^2 + (dz/dy)^2) dA
where dA is the area element of S.
We can simplify the integral by using the fact that dz/dx = -3/12 = -1/4 and dz/dy = -4/12 = -1/3:
∫∫S √(1 + (dz/dx)^2 + (dz/dy)^2) dA
= ∫∫S √(1 + 1/16 + 1/9) dA
= ∫∫S √(325/144) dA
= (5/12) ∫∫S dA
The integral ∫∫S dA is the area of the region enclosed by the cylinder and the plane. We can find this area by integrating over the projection of the region onto the xy-plane. The projection is the region between the parabolas y = 4x^2 and y = 8 - 4x^2. We can integrate this region using polar coordinates, with the limits for r from 0 to (2/y)^(1/2) and the limits for θ from 0 to π:
∫∫S dA = ∫₀^π ∫₀^(2/y)^(1/2) r dr dθ
= π ∫₀^∞ (2/y)^(1/2) r dr
= π (2/y)^(1/2) ∫₀^∞ r dr
= π (2/y)^(1/2) (1/2) [r^2]₀^∞
= π (2/y)
Substituting this into the surface area integral, we get:
Surface area of S = (5/12) ∫∫S dA = (5/12) π (2/y) = (5/6) π (3x + 4y -
Explanation: