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Introduction | Exercise 04[Volume of Solids]|

Volume of Solids

Volume by Double Integration

 Consider a surface . Let the orthogonal projection on XY plane of its portion S’ be the surface S. Divide S into elementary rectangles of area  by lines parallel to x and y axis. Now consider prism have its base as these triangles and its length parallel to OZ. The volume of this prism under consideration is bounded by surface S and surface  is . Hence the volume of the solid cylinder having S as base, bounded by the given surfaces with generator parallel to Z-axis is Where integration is to carried over surface S. In polar coordinates the above integration becomes Example: Lets us find the volume bounded by the cylinder  and the planes  and . Here, S’ i.e. is the plane , and its projection i.e. S on  will also be a plane surrounded by the base of the cylinder . Since the top plane is symmetrical about , so we first find half the volume and will double later to find actual volume. Hence limits for the projected surface i.e.  are as: x varies from 0 to , and y varies from -2 to 2. So, required Volume is