A vertical fence is constructed whose base is the curve y= x* sqrt(x) from (0,0) to (1,1) and whose height above each point (x,y) along the curve is x^3 - y^2 + 27. Find the area of this fence.

The correct answer given by the book is D. 13*sqrt(13) - 8, which is obtained by solving a path integral.

However I was a bit confused by this and I thought that a path integral in this case wouldn't give you the "area" of the fence but something else.

I thought the height above each point on the curve is simply 27 by plugging in y=x*sqrt(x) and so the area of the fence should simply be obtained by integrating 27 + x*sqrt(x) - x*sqrt(x) = 27 from 0 to 1.

So my answer would be E. 27

Any thoughts on this?

Thanks!