Hi, we're currently covering 3 dimensional slicing with integrals in my calculus class, which is highly intuitive stuff. We've been calculating volumes via integrating "slices" of the figures in question.
I brought up whether something similar could be done for the surface area of curved objects, and my teacher told me surprisingly that it can't be done.
I'm not entirely sure that's correct; it seems like it should work. If you can slice a sphere into infinitely many theoretical circles, why can't you slice its surface area into infinitely many planar figures? To make it simpler, I'll go with a cone: the curved portion of a right circular cone's surface area can be thought of as a series of infinitesimally small equilateral triangles mutually bounding each other and all converging on a single point. If one of these is used as the tail end of an integral (change in x, y, whatever), can it be done?
Or alternately, in the way that the areas of infinitely many circles are used to calculate the volume of the cone, can't you integrate (using the same bounds) with respect to the circumferences of these circles rather than their areas and get the surface area?
I brought up whether something similar could be done for the surface area of curved objects, and my teacher told me surprisingly that it can't be done.
I'm not entirely sure that's correct; it seems like it should work. If you can slice a sphere into infinitely many theoretical circles, why can't you slice its surface area into infinitely many planar figures? To make it simpler, I'll go with a cone: the curved portion of a right circular cone's surface area can be thought of as a series of infinitesimally small equilateral triangles mutually bounding each other and all converging on a single point. If one of these is used as the tail end of an integral (change in x, y, whatever), can it be done?
Or alternately, in the way that the areas of infinitely many circles are used to calculate the volume of the cone, can't you integrate (using the same bounds) with respect to the circumferences of these circles rather than their areas and get the surface area?