Area between curves

Question

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.

Here is what I did..

So we have two curves..the one on the top can be written in this form $$y=x^3-(x\sqrt x)^2+27+x\sqrt x=27+x\sqrt x$$ And the curve below is $x\sqrt x$..So the area between them would be given by $$\int 27+x\sqrt x\,dx-\int x\sqrt x\,dx$$ over $x$$=0.1$ which gives us $27$.

But this doesn't match with the solution. Please take a look.

Answer 1

My hunch is that you are not meant to be looking at two curves in the $x,y$ plane. I think you only have one curve in that plane. All the rest of your figure is "above" the plane; think of a curved sheet of something that sticks up in the $z$ direction in three-dimensional space.

So what you want is the area of a curved surface that lies between the curve $y=x\sqrt x, z=0$ and the curve $y=x\sqrt x, z=x^3-y^2+27$ in three-dimensional space.

Step number one, see if you can figure out the values of $z$ on the second curve.

You probably need to look at the formulas for length of a curve in a plane. Consider the curved sheet as composed of a lot of skinny rectangles, each one having one very short edge on the curve on the $x,y$ plane and the other edge touching the curve above.

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Addendum: It might pay to look at other problems and examples given by the writer of this problem (that is, if this is a problem from a textbook or a textbook's online exercises, read the book) in order to look for clues about what the author could have meant by "vertical", "fence", "base", and "height".

It seems likely that the "fence" is likely to be a set of parallel line segments, but does the author mean "vertical" in the sense of the vertical line test for a single-variable function plotted in the $x,y$ plane (so "vertical" is parallel to the $y$-axis), or is "vertical" perpendicular to the plane in which the "base" curve lies (so "vertical" is parallel to the $z$-axis). If "vertical" is parallel to the $y$-axis, does "height" mean the length of each parallel line segment (probably not the intended interpretation, since I also get $27$ as the area in that case) or does "height" mean the $y$-coordinate of the "upper" end of the line?

Perhaps your instructor can clarify the meaning of the words in this question.

If there really are no good clues about what a "vertical fence ... whose base is [a curve]" is and you cannot get a clarification of them, I am sorry you are stuck answering questions from someone who requires such guesswork to answer their questions.