Let a b c and d be positive real numbers The first octant of

Let a, b, c, and d be positive real numbers. The first octant of the plane ax + by + cz = d is shown in the figure. Show that the surface area of this portion of the plane is equal to where A(R) is the area of the triangular region R in the xy-plane, as shown in the figure.

Solution

Herewith the steps. 1. Solve the equation for z and write it as f(x,y). 2. Calculate partial of f with respect to x to get -a/c and then calculate partial of f with respect to y to get -b/c. 3. Now use the double integral formula for finding surface area. This is \'double integral over the region R\' where the integrand is square root of \'f w.r.t x squared + f w.r.t y squared + 1.\' 4. Substitution will now give you Surface area = double integral over R of \'square root of (a^2/c^2 + b^2/c^2 + 1)\' 5. Factor out this constant (and factor out the \'over c^2\' from the square root). 6. Now you have the constant out front - square root of a^2 + b^2 + c^2 all over c - and double integral dA but this exactly the A(R) that they gave you in the problem.