Page:Calculus Made Easy.pdf/227

 beneath the piece $$PQ$$. The problem is, how can we calculate the value of this area?

The secret of solving this problem is to conceive the area as being divided up into a lot of narrow strips, each of them being of the width $$dx$$. The smaller we take $$dx$$, the more of them there will be between $$x_1$$ and $$x_2$$. Now, the whole area is clearly equal to the sum of the areas of all such strips. Our business will then be to discover an expression for the area of any one narrow strip, and to integrate it so as to add together all the strips. Now think of any one of the strips.

It will be like this: being bounded between two vertical sides, with a flat bottom $$dx$$, and with a slightly curved sloping top. Suppose we take its average height as being $$y$$; then, as its width is $$dx$$, its area will be $$y\, dx$$. And seeing that we may take the width as narrow as we please, if we only take it narrow enough its average height will be the same as the height at the middle of it. Now let us call the unknown value of the whole area $$S$$, meaning surface. The area of one strip will be simply a bit of the whole area, and may therefore be called $$dS$$. So we may write

If then we add up all the strips, we get

So then our finding $$S$$ depends on whether we can