Page:Calculus Made Easy.pdf/239

 Example. The curve $$y=x^2-5$$ is revolving about the axis of $$x$$. Find the area of the surface generated by the curve between $$x=0$$ and $$x=6$$.

A point on the curve, the ordinate of which is $$y$$, describes a circumference of length $$2\pi y$$, and a narrow belt of the surface, of width $$dx$$, corresponding to this point, has for area $$2\pi y\, dx$$. The total area is

When the equation of the boundary of an area is given as a function of the distance $$r$$ of a point of it from a fixed point $$O$$ (see Fig. 61) called the pole, and



of the angle which $$r$$ makes with the positive horizontal direction $$OX$$, the process just explained can be applied just as easily, with a small modification. Instead of a strip of area, we consider a small triangle $$OAB$$, the angle at $$O$$ being $$d\theta$$, and we find the sum