In this case too, cross section would be perpendicular to axis of rotation. Since the area of washer (ring) is given as so we are going to use this concept to find volume of solid using this method. If we obtain the solid of revolution with a hole inside it, then we use washer method because the cross section is in form of a washer. Since area is symmetrical about x axis so we can use property of definite integrals. So we set both equations equal and solve. To find limits of integration we find intersecting points of this parabolic curve with line x=3. Next important thing is to find radius of this circular disk. Since axis of revolution is a vertical line(x=3) so cross section would be horizontal having an approximating disk of width dy. Cross section is shown with green slice(disk) which is perpendicular to axis of rotation(x axis) having width dx. Solution: Here region R given by y=√x is being rotated about x axis and we get resulting solid shape as in (b). When region y=f(x) is rotated about x axis: Then cross section is vertical (perpendicular to x axis) and approximating rectangle of cross section has width dx.Įxample1: Find the volume of the solid obtained by revolving the region under the graph of y=√x on about the x-axis. In this method cross section is taken perpendicular to axis of revolution. In this method when a curve y= f(x) defined and continuous on is rotated about x or y axis, then a solid shape is obtained without any hole at the center. Volume of solids of revolution can be found using following three ways: Solid of revolution is generated by revolving a plane curve about x or y axis or about any line that lie in the same plane as the given function.
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