Find the surface area of the portion of the paraboloid z = 4 - 𝑥^2 -𝑦^2 that lies above the xy- plane?
should have tried google first.
http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/surface/surface.html
has this exact problem, only the circle is or radius 4, rather than 2.
To find the surface area of the portion of the paraboloid that lies above the xy-plane, we need to calculate the surface area of the curved surface.
The equation given represents a paraboloid with its vertex at (0, 0, 4) and opening downwards. The xy-plane is the plane where z = 0.
To find the surface area, we can use the formula for the surface area of a general surface given by a parameterized form:
A = ∬√((∂z/∂x)^2 + (∂z/∂y)^2 + 1) dA
where ∂z/∂x and ∂z/∂y are the partial derivatives of z with respect to x and y, and dA represents the differential area element in the xy-plane.
In this case, we need to parameterize the surface in terms of two variables, x and y.
Let's express z in terms of x and y:
z = 4 - x^2 - y^2
Next, we need to find the partial derivatives of z with respect to x and y:
∂z/∂x = -2x
∂z/∂y = -2y
Now, we can substitute these values into the surface area formula:
A = ∬√((-2x)^2 + (-2y)^2 + 1) dA
To evaluate this double integral, we need to determine the region of integration on the xy-plane. Since we want the portion of the paraboloid that lies above the xy-plane, the region of integration is the entire xy-plane.
The dA element represents a small differential area on the xy-plane. Since we are integrating over the entire xy-plane, we can use standard Cartesian coordinates for the region of integration.
The limits of integration for x and y will be from negative infinity to positive infinity.
Finally, we integrate the expression to find the surface area:
A = ∫∫√((-2x)^2 + (-2y)^2 + 1) dx dy
However, this integral does not have a simple closed-form solution. It may require numerical methods or approximation techniques to find an estimate of the surface area.
Alternatively, if you have a specific region of integration on the xy-plane in mind, you can provide those limits, and we can calculate the surface area for that specific region.