To find the area of the surface of the plane defined by the equation x + 2y + 3z = 1 that lies inside the cylinder defined by x² + y² = 3, we can follow these steps:
1. **Express z in terms of x and y**: The equation of the plane can be rearranged to express z:
z = (1 – x – 2y) / 3
2. **Find the region of integration**: The cylinder x² + y² = 3 defines a region in the xy-plane. We’ll convert it into polar coordinates where x = r cos(θ) and y = r sin(θ), leading to the bounds r² = 3 or r = √3 for 0 ≤ θ < 2π.
3. **Calculate the surface area element**: The surface area A can be found using the formula:
A = ∬_D √(1 + (∂z/∂x)² + (∂z/∂y)²) dA
where D is the projection of our surface onto the xy-plane.
4. **Calculate the partial derivatives**:
We need to calculate the partial derivatives ∂z/∂x and ∂z/∂y:
∂z/∂x = -1/3 and ∂z/∂y = -2/3.
Now, substitute these into the formula:
A = ∬_D √(1 + (-1/3)² + (-2/3)²) dA
Simplifying gives:
A = ∬_D √(1 + 1/9 + 4/9) dA
A = ∬_D √(1 + 5/9) dA
A = ∬_D √(14/9) dA = (√14/3) ∬_D dA
5. **Evaluate the area element**: The area element in polar coordinates is dA = r dr dθ. The limits for r are from 0 to √3 and for θ from 0 to 2π:
A = (√14/3) ∫(0 to 2π) ∫(0 to √3) r dr dθ
6. **Calculate the integrals**:
First evaluate the inner integral:
∫(0 to √3) r dr = [r²/2](0 to √3) = (3/2)
Now evaluate the outer integral:
A = (√14/3) ∫(0 to 2π) (3/2) dθ = (√14/3) * (3/2) * (2π) = √14 * π
So, the total area of the surface of the plane that lies inside the cylinder is A = √14 * π.