To find the surface area of the paraboloid defined by the equation y = x² + z² that is constrained within the cylinder defined by x² + z² = 9, we need to follow a few steps involving calculus.
First, we recognize that the region we are interested in is a circular disk in the xz-plane with radius 3, since x² + z² = 9 represents a circle. The limits for x and z will therefore be determined by this circle.
Next, we will set up the surface area integral using the formula for surface area, which is given by:
A = ∬_D √(1 + (∂f/∂x)² + (∂f/∂z)²) dAIn our case, f(x, z) = x² + z². We need to calculate the partial derivatives:
- ∂f/∂x = 2x
- ∂f/∂z = 2z
Now we can substitute these derivatives into the surface area formula:
A = ∬_D √(1 + (2x)² + (2z)²) dA = ∬_D √(1 + 4x² + 4z²) dANext, we switch to polar coordinates to simplify the region of integration. Setting:
- x = r cos(θ)
- z = r sin(θ)
where r² = x² + z² and allows us to integrate over a circular region.
The Jacobian of the transformation is r, so the area element in polar coordinates becomes dA = r dr dθ.
Now our limits for r will be from 0 to 3 (since the radius of the cylinder is 3) and for θ from 0 to 2π:
A = ∫(from 0 to 2π) ∫(from 0 to 3) √(1 + 4r²) r dr dθThis double integral can now be evaluated by first integrating with respect to r and then with respect to θ.
Finally, after evaluating the integral, we obtain the total surface area of the paraboloid constrained within the cylinder. This would yield the required surface area.