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SERGE LANG. The simplest case was that of a pyramid with square base, or rectangular base, and a given height. Now suppose you have an arbitrary curved base. How can you deal with an arbitrary curved base? What do I do? SERGE. Oh, I see. You try to fit other figures by means of a grid. SHERYL. You approximate the base with a grid. SERGE LANG. I put a rectangular grid on the base, that's right. Then start drawing all these lines r VOLUMES IN HIGHER DIMENSION 49 Vertex RACHEL. You approximate the volume by approximating the base with rectangles.
You, what's your name? STUDENT. Christopher. SERGE LANG. All right, Christopher. So? NATHALIE. The circumference of a circle. SERGE LANG. So what is the circumference of a circle? NATHALIE. What do you mean, what is it? SERGE LANG. I mean, as a function of '11'. NATHALIE. THE VOLUME OF THE BALL 53 SERGE LANG. Well, you have a circle of radius r ... A STUDENT. Ah! It's 2'lTr. SERGE LANG. Very good. 2'lTr, where r is the radius. And the surface, the area, what is it? THE STUDENT. It's 'lTr squared.
Then we can make a dilation by a factor of a on one side of the base, a factor of b on another side of the base, and a factor of h vertically, and you get the formula for your type of pyramid, when the vertex lies directly above one corner of the base: 44 VOLUMES IN HIGHER DIMENSION ~ \ Volume \ \ \ \ h \ \ = 13 abh = 13 Bh \ )/ / / a But then you still have the problem of slanting to deal with. So let's do what Serge wants to do. Suppose I have a rectangle for the base, and suppose I have a slanted pyramid, as on the picture.