Question

An artificially created lake is bordered on one side by a straight dam. The shape of the lake surface is that of a region in the xy-plane bounded by the graphs of 2y=16-x^2 and x+2y=4. Find the area of the surface of the lake.

Answer 1

This is just like the last one, but oriented the other way. The graphs intersect at x = -3 and 4.

So, the area is just

∫[-3,4] f(x)-g(x) dx
where
f(x) = (16-x^2)/2
g(x) = (4-x)/2

Answer 2

Here is the area you are finding
http://www.wolframalpha.com/input/?i=plot+2y%3D16-x%5E2+%2C+x%2B2y%3D4

Find the intersection of the two graphs, you only really need the x values.

take vertical slices.
so the height of a slice is (8-x^2/2) - (2 - x/2)
= 6 - x^2/2 + x/2

take the definite integral of that from the left x to the right x of your intersection points